Project: ArangoDB (App Definition and Development / Database)
File: TQE.pdf

The Quaternionic Exponential
(and beyond)
Hubert HOLIN
Hubert.Holin@Bigfoot.com
http://www.bigfoot.com/~Hubert.Holin
08/12/1999
Motivation ............................................................................................................................... .......2
Chapter 1  Quaternions redux .....................................................................................................2
1- What to find here .....................................................................................................................2
2- The nature of the Beast ...........................................................................................................2
3- Quaternions’ kin............................................................................................................ ..........4
4- Quaternions and rotations................................................................................................... ...5
5- Miscellany ............................................................................................................................... ..8
Chapter 2  Building the Quaternions .......................................................................................1 1
1- What to find here ...................................................................................................................1 1
2- Cayley algebra, alternative algebra...................................................................................1 13- The Cayley doubling procedure..........................................................................................1 24- 
R, C, H, O, X.......................................................................................................................1 3
5- The full Cayley ladder all at once ........................................................................................1 4
Chapter 3  The Exponential .......................................................................................................1 8
1- What to find here ...................................................................................................................1 8
2- Definition ............................................................................................................................... .1 8
3- Links with differentiation.................................................................................................. .1 8
4- The closed formula for the exponential in   CR().............................................................1 9
5- Some properties of the exponential and further consequences..................................2 06- Conclusion ............................................................................................................................... 20
Bibliography ............................................................................................................................... 21
Software index .............................................................................................................................2 1
1Interesting URLs .............................................................................................................. .........2 1Motivation
I felt the need to take a closer look at quaternions when, some time back, I was
looking for new applications to Harthong-Reeb circles (on which I was working at thetime), and came across [
D. Pletincks (1989) ]. That paper, on one hand, did indicate one
potential application for that method, but, on the other hand, alluded to some oddconstructions involving quaternions, the validity of which was propitiously left in theshadows. The present text is therefore a compilation of many well-known but apparentlyscattered results about quaternions (and related entities), as well as some newdevelopments, notably the explicit formula for the quaternionic exponential (and friends).Incidentally, these results enables one to solve the problem found in [
D. Pletincks
(1989)], but without the unsalvageable constructions.
Chapter 1  Quaternions redux
1- What to find here
This chapter only contains a quick-and-dirty (but sufficient for most uses)
presentation of the quaternions, along with their most classical properties, inspiredvery largely by [
D. Leborgne (1982)], [J. Lelong-Ferrand, J.M. Arnaudiès (1978)] and [M. Berger
(1990)]. This approach, however, obscures the deep relationship which links the
quaternions, the complex and real numbers and more exotic things known as octonions;this relationship will be the thrust of the next chapter.
It should be said that other important uses of quaternions exist ([
K. Gürlebeck,
W. Spössig (1989)],...), but that they will not be touched upon here. As well, quaternionic
analysis ([ A. Sudbery (1979)]) and geometry ([ S. Salamon (1982)]), though perhaps not as
vibrant as their complex counterparts, do keep evolving; though these usually involvefairly sophisticated mathematical machinery, very nice results can also be had withvery elementary ones ([
P. de Casteljau (1987)],...). All are beyond the scope of this article,
however.
2- The nature of the Beast
Let H=R4 with the usual four-dimensional vector space structure over R. We
define e=()1000,,,, i=()0100,,,, j=()0010,,, and k=()0001,,,.
The first important thing we need is  a multiplication, denoted *, which we
define to be a (non-commutative) R-bilinear operation on H such that ii jj kk e*=*=*= - ,
ij ji k*= -*()=, jk kj i*= -*()= and ki ik j*= -*()=.
The second important thing we need is the conjugation  on H (and we will
usually denote by q the conjugate of q) which we define by   abgd a b g d,,, , , ,() ---()a .
Important properties are that qq q q*¢=¢*, that ee=, that qq qq*=*˛× Re and that
qq+˛×Re. Actually qq*=0 if and only if q=0, as is easily seen.
A straightforward verification then shows that H,,,+*×() is an effectively non-
commutative, but associative, R-algebra with unit e, and that   RHﬁ () [], ,,,xxa000 and
  CHﬁ () ()() [],R e , I m , ,zz za 00 are algebra homomorphisms, bijective from their sources
2onto their images. The image of the conjugate of a complex number is also seen to bethe conjugate (in H) of the image of that complex, by the above function. We will
therefore assimilate H to a superset of (both) R and C, and identify e with 1 and i
with its complex counterpart. We see at once that the operations we have defined on
H extend their counterparts on C and R. The multiplication can then be memorized
thru the well-known formula:
ii jj kk i k*=*=*=**= - j1
It is important to notice that given any quaternion q and any real number x, we
always have qx xqx q*=*=× .
We will usually write a quaternion under the form q=+ + +ab g dijk  with a, b,
g and d reals, omitting the “ ×” when multiplying a quaternion by a real number (as per
the vector space structure). We will also omit the “ *” when multiplying a quaternion
by a real number, from the left as well as from the right. When no confusion mayarise, we will do away entirely with the “
*”.
With the above notations, the conjugate of q=+ + +ab g dijk  will then simply be
q=- - -ab g dijk .
Looking at H as a 4-dimensional R-vector space, it is easy to see the usual
euclidian scalar product is equal to the following:
p q pqpq p pq q
pq qp
pq qp()=+() +()--
=+()
=+()1
2
1
2
All the same, the usual euclidian norm on R4, coincides with   qq q qa=*[] ,
and of course qq q()== + + +22222abgd . Note that, if q„0 then qq q q q q q- --=*() =*()1 11.
For the quaternions, we will also use a notation compatible with real and complex
numbers and define q as q (of course, if q is actually complex, q has exactly the
value of the modulus of q).
It is important to remember that H,,, ,+*×()  is a Banach R-algebra. The norm is
better than what we might expect, though, as we have pq p q*= instead of just
pq p q*£.
We will call the real and unreal parts of quaternion, respectively, Reqq q()=+()1
2
and Urqq q()=-()1
2. We will say that a quaternion is pure if its real part is zero. For a
complex number, the quaternionic real part is what is already known as the complex
real part, and the unreal part is just the imaginary part multiplied by i.
33- Quaternions’ kin
As we have just seen, quaternions are related to both real numbers and complex
numbers. As we shall see in some details in the next chapter,  quaternions areactually part of an infinite family of sets
1 which we will call the Cayley ladder, some
of which we will introduce here as we will have some need of them for our purposes.
First relative in that family, beyond the quaternions, are the octonions. We
denote by O the set R8, with its usual vector space structure on R, we identify
1 10000000=(),,,,,,, , i=()01000000,,,,,,, , j=()00100000,,,,,,,  and k=()00010000,,,,,,,  and we define
¢=()e00001000,,,,,,, , ¢=()i00000100,,,,,,, , ¢=()j00000010,,,,,,,  and ¢=()k00000001,,,,,,, . We now
consider O to be a super-set of H. We can now define a multiplication on O by the
following table (the value at line n and column m is the product of the element in the
left column by the element in the top row; for instance ii e*¢=-¢):
1
11
1
1
1
1
1ijk eijk
ijk eijk
ii k j iek j
jjk ijkei
kkj i k ji e
ee ijk ijk
iiek j i k j
jj¢¢¢¢
¢¢¢¢
-- ¢-¢-¢¢
-- ¢¢ -¢-¢
-- ¢-¢¢-¢
¢¢ -¢-¢-¢-
¢¢¢ -¢¢ ---
¢¢¢ kke i j k i
kk ji e k ji¢-¢-- -
¢¢ -¢¢ ¢ -- -1
1
Other presentations, perhaps more useful, exist ([ G. Dixon]). This multiplication still
has a unit ( 1), but is no longer associative (for instance ¢*¢*()=+ „- = ¢*¢()* iej k ki ej ).
Real numbers still commute with every octonion. We define a conjugation by
ab g d e z h q ab g d e z h q++++ ¢+¢+¢+¢=- - - - ¢-¢-¢-¢ ijkei jk ijkei jk , a scalar product and
a norm which, as with the quaternions turn out to be exactly the euclidian scalarproduct and euclidian norm on 
R8. Again, we have just extended the quaternionic
operations. As with complex numbers and quaternions, we have oo o o*¢=¢ for any two
octonions o and ¢o, and an octonion o is invertible if and only if it is non-zero, and
then ooo-=1
21.
Beyond even the octonions, we find R16, which appears not to have any agreed-upon
name. We shall here call them hexadecimalions, and denote the set by X (after the
C/C++ notation...). We have the usual vector space structure on R, we identify   1, ,K¢k
with   1000000000000000 0000000100000000,,,,,,,,,,,,,,, , , ,,,,,,,,,,,,,,,() () K  respectively, and define
¢¢ ¢¢ ¢¢ ¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢eijkei j k,,,, ,, ,  as   0000000010000000 0000000000000001,,,,,,,,,,,,,,, , , ,,,,,,,,,,,,,,,() () K
respectively. We define a multiplication on  X as explicited in the next chapter, for
41Actually, several families, but we will focus on just one here; for others, see [G. Dixon (1994)].which 1 is still a unit and for which reals commute with every hexadecimalion. Wedefine as well a conjugation, a scalar product and a norm (for details, see next
chapter), which once again coincide with the euclidian scalar product and euclidiannorm on 
R16. These all extend the octonionic case. However, the product has even
fewer properties than in the octonionic case (the algebra is no longer even alternative2,
as for instance ie ie j j k j ie ie j+¢¢¢()*+¢¢¢()*() =- + „- = + ¢¢¢()*+¢¢¢()() * 22 2 ), and the norm is
not even an algebra norm any longer, as for instance
ij ek ij ek+¢¢()*¢+¢¢¢() =>=+ ¢¢ ¢+¢¢¢2 2284 .
4- Quaternions and rotations
It is pleasant to think that perhaps the relationship between quaternions and
rotations has been stumbled upon while running a check-list of classical constructson the then-newly discovered quaternions. At any rate, the easiest way to explain thatlink is thru interior automorphisms.
More precisely, given a non-zero  quaternion 
q=++ +abg dijk , we can consider
the interior automorphism:
  lq
pq p q:HHﬁ
()( )-a1
These objects have several fundamental properties:   ll lqq qq ¢¢=o  and lqqq()=, lq
leaves R invariant (since reals commute with all quaternions), and lq respects the
norm on H.
It is interesting to see lq as an R-linear function on H. As it preserves the
norm, it preserves the scalar product, and hence lq˛()O4 ,H. Then, as it leaves R
globally invariant, it must leave its orthogonal ( i.e. the unreals) globally invariant.
Consider now the matrix of lq ; expressed in the canonical basis   C=()1, , ,ijk that
matrix is:
  MC Clabgd
a b g d ad bg ag bd
ad bg a b g d ab gd
ag bd ab gd a b g dqq,,() =+++
+-- - + + +
++ - + - -+
-+ ++ - - +Ø
ºŒ
ŒŒŒ1000
02 2 2 202 2 2 202 2 2 2
22222
2222
2222
2222øø
ßœ
œœœ
It is quite obvious3 that     QM C C:, , ; , ,R404 4-{}ﬁ() () [] MRqq al is continuous, and a
group homomorphism. As we have seen,   QRR404-{}() Ì()O,, and as   Q14()=I, the identity
52We will define this in the next chapter.
3We will note MUnm,,() the set of matrices, n rows by m columns, with elements in U.
4More generally, we will denote by In the identity matrix on Rn.matrix4 on HR=4,   QR40-{}() must actually be included in the connected component ofI4 in O4 ,R(), and that is SO 4,R(), i.e., lq is a rotation on R4, and hence on R, where it is
the identity I1, and thus must also be a rotation on 03{}·R, i.e. the unreals. We can find
a far simpler (if somewhat tedious) proof of that by simply computing the determinant
of   MC Clq,,(), which of course turns out to be 1 (also see next section)...
We can therefore extract a rotation matrix on R3 from   MC Clq,,():
rabgda b g d ad bg ag bd
ad bg a b g d ab gd
ag bd ab gd a b g dq=++++-- - + + +
++ - + - -+
-+ ++ - - +Ø
ºŒ
ŒŒø
ßœ
œœ122 22
22 22
22 22
22222222
2222
2222
Let us introduce     R:, , ;R403 3-{}ﬁ() [] MRqqar. It is trivial to see that   Q and   R
are both C¥ (because they are rational). It is important to note that they are both
R-homogeneous of degree 0, which means that given any non-zero real number x, lq
and lxq are identical, and therefore yield identical rotations ( i.e. rrqx q=).
A fundamental result is that   R is surjective. There are at least two well-known
ways to prove this.
The easiest way also has the advantage of being completely constructive: we
just compute the elements of the rotation rq.
This is possible because we always know one invariant vector. Indeed (as an
immediate consequence of lqqq()=):
rb
g
db
g
dqØ
ºŒ
ŒŒø
ßœ
œœ=Ø
ºŒ
ŒŒø
ßœ
œœ
Furthermore, the angle, qp˛[]0;, is given by considering the trace of rq:
1232222
2222+()=---
+++cosqabgd
abgd
We now exploit the homogeneity of   R, which implies that   RRHS-{}() =()03, and
therefore that we can restrict our search to unit quaternions. For unit quaternions,
the trace relation simplifies to 122+()=cosqa.
Therefore, the identity rotation I3 is associated with q=–1 (which we already
knew), and these unit quaternions only.
Let   rrr
ijk,,() be the canonical basis of R3. Consider now a rotation r„I3 (hence
qp˛]]0;), it possesses a unique rotation axis, and a unique unit vector   rrr r
ux iy jz k=++
directing that axis such that   ra uau u a arr r r r r r()=-()() ×()+()Ù()+() 1 cos sin cosqq q  for all   ra˛R3. It
6follows that r is associated with the two unit quaternionsqx
y
z=–æ
Łö
ł
æ
Łö
ł
æ
Łö
ł
æ
Łö
łØ
ºŒ
ŒŒŒŒŒŒŒŒŒŒŒø
ßœ
œœœœœœœœœœœcos
sinsinsin
q
q
q
q2
222
and these two unit quaternions only.
The second method is non-constructive, but has the advantage of highlighting
the regularity of the connection between rotations and quaternions, which is harder toread using the first method.
We once again exploit the homogeneity of 
  R and use unit quaternions. Given
that we know that in fact   RSR3()Ì()SO 3,, we can consider   RSR
3SO 3,() which is C¥ (because
it is rational). It is slightly tedious, but possible, to prove that in fact   RSR
3SO 3,() is a
local diffeomorphism at 1. It is also a group homomorphism (stemming from the fact
that   ll lqq qq ¢¢=o). Since in a connected topological group, every neighborhood of the
neutral element is a generator of the whole group ([ G. Pichon (1973), p 31]),   RSR
3SO 3,() is
surjective upon the connected component of   I31=()R in SO 3,R(), i.e. upon SO 3,R(), and
of course is everywhere a local diffeomorphism (though it is of course not a global
diffeomorphism).
Combining these two approaches, one finds a global C¥-diffeomorphism between
SO 3,R() and RP3 (which is nothing more than S3 where every couple of opposite points
have been identified).
Another thing worth noting is that   RSR
3SO 3,() is more than just a locally diffeomorphic
bijection. If we call sS3 the positive Borel measure on S3 induced by HR=4 and sSO 3,R()
that induced on SO 3,R() by MR,,33() (by assimilation of the rotations with their matrix
in the canonical basis of R3), seen as R9, then we can compute5 that   R*ssSO 3,R S ()=16 23.
Furthermore,   RSR
3SO 3,() actually has no critical point.
75 A fact that is supposed to be found, but is not, in [C.W. Misner, K.S. Thone, J.A. Wheeler (1973)].5- Miscellany
As we have seen, the main power of the quaternions is their ability to pleasantly
parameter SO 3,R(). It should be said that what is, perhaps their greatest strengths in
this regard, with respect to other parameterization of SO 3,R() such as Euler angles, is
that   RSR
3SO 3,() has no critical points (no “Gimbal Lock”), and that the composition of
rotations is extremely simple to compute in terms of the parameter. Also and they
can be shown to allow interpolations of orientations under constraints (such hashaving one axis stay “horizontal”).
Quaternions also allow a nice parameterization of 
SO 4,R() ([M. Berger (1990)] the
application   SS R334 ·ﬁ () ( ) [] SO , , , sr q s q raa  is a continuous group homomorphism,
surjective, with kernel 11 1 1,, ,()--(){} ).
Quaternions have other uses, though. For instance, they can be also be used to
parameter SU 2,C(). More precisely, an isomorphism exists between 03{}·S and SU 2,C()
(consider, the application
  Y:, , HC ﬁ()
=+++-Ø
ºŒø
ßœM2 2
qi j kuv
vuabgd a
with ui=+ad and vi=+gb is a ring isomorphism from H on a sub-ring of MC,,22(),
which induces an isomorphism). There are also applications of quaternions to the
Riemann sphere ([ J. Lelong-Ferrand, J.M. Arnaudiès (1978)]).
It should be mentioned that research exists to find more efficient algorithm
for the product of quaternions ([ T. Howell,  J.C. Lafon (1975)]), but has so far not reached a
conclusion, one way or the other.
Given the power of the quaternions, the question naturally arises as to whether
something similar can be done for rotations on spaces of higher dimensions (themultiplication being commutative on the reals and complex numbers, interiorsautomorphisms are just the identity). The answer to that question is partly positive,but it should be now stated that the right tool, in general, for that problem turns outto be Clifford algebras rather than Cayley algebras.
When we turn to the octonions, the multiplication is not only not associative, it
is no longer even associative. Fortunately, the sub-algebra engendered by any twoelements (and the unity) is still associative, and therefore interior automorphism donot depend on the order in which the products are carried out. The interesting fact isthat, as with the quaternions, the interior automorphisms leave 
R invariant, and
induce a rotation, on R7 this time. The catch is that SO 7,R() is a 21-dimentional
manifold, whereas the interior automorphisms we just described only have 7 degreesof freedom. In short, we do not get all the rotations on 
R7 by this method. It is still
useful, though, for theoretical purposes.
8Beyond the even the octonions, the hexadecimalions have two different flavors
of interior automorphism,   pq p qa()()()-1 and   pq p qa()()()-1, neither of which is, in
general, a rotation (on either R16 or R15). The average of the two isn’t a rotation
either, by the way...
Interior automorphisms having apparently reached the limits of their usefulness,
we turn now to something else, with the same objects. It turns out that we can findrotations with even simpler constructions!
Let 
x=˛aR, then     Mxyx y yy x=[]() =[]() =[] MM aa,, ,,11 11 a, hence tMM Ixxx=1, and
det Mxx()=. Therefore if x=1, we find that MOx˛()1,R, and we of course get all two
elements of O1 ,R() that way... but MS Ox˛()1,R only if x=1! Obviously, given x˛R and
¢˛xR, MM M M Mxx xxx x ¢¢ ¢== .
Let ci=+ ˛abC, then 
    Mczc zii zz cii=[]() ()() =[]() ()() =-
+Ø
ºŒø
ßœMM aa,,,, ,,,,11 11ab
ba, hence
tMM Iccc=2, and det Mc()=+ab22. Therefore if c=1, MS Oc˛()2,R, and we get all rotations
on R2 that way, as is well-known. And given c˛C and ¢˛cC we still have
MM M M Mcc ccc c ¢¢ ¢== .
Let now q=++ + ˛abg dijk H, then:
    MqGpq pi j ki j k=[]() ()() =---
+- +
++ -
+-+Ø
ºŒ
ŒŒŒø
ßœ
œœœM a,, ,,,, ,,11
ab gd
ba d g
gd ab
dgb a
and
    MqDpp qi j ki j k=[]() ()() =---
++ -
+- +
++-Ø
ºŒ
ŒŒŒø
ßœ
œœœM a,, ,,,, ,,11
ab gd
ba d g
gd ab
dgb a
, hence ttMM MM IqG
qG
qD
qDq ==4, and det detMMqG
qD()=()=+ + +()abgd22222. Therefore if q=1,
MS OqG˛()4,R and MS OqD˛()4,R, but we only get a tiny fraction of SO 4,R() that way.
This, of course can be used as an alternate proof that the interior automorphisms
on the quaternions actually induce rotations on R4.
It is interesting to note that given q˛H and ¢˛qH, we still have MM MqqG
qG
qG
¢¢=
9and MM MqqD
qD
qD
¢¢=, though we now sometimes have MM MMqG
qG
qG
qG
¢¢„ and MM MM¢¢„qD
qD
qD
qD.Turning to the octonions, let oi j k e i j k=+ + + + ¢+¢+¢+¢˛ ab g d e z h q O, then:
    MoGoo oi j k e i j ki j k e i j k=¢¢[] ¢¢¢¢() ¢¢¢¢() () =-------
+ -+-++-
++ ---++
+-+ -+-M a,, ,,,,,, ,, ,,,,,,11ab gde zh q
ba d g z e q h
gd abhqez
dgb aqhz ze
ezhq a bgd
zeqhb adg
hqezgda b
qhzedgb a+
++++ ---
+-+-+ +-
+--++- +
++--++-Ø
ºŒ
ŒŒŒŒŒŒŒŒŒ
Œø
ßœ
œœœœœœœœœ
œ
and
    MoDoo oi j k e i j ki j k e i j k=¢¢[] ¢¢¢¢() ¢¢¢¢() () =-------
+ +-+--+
+- +++--
++- +-+M a,, ,,,,,, ,, ,,,,,,11ab gde zh q
ba d g z e q h
gd abhqez
dgb aqhz ze
ezhq a bgd
zeqhb adg
hqezgda b
qhzedgb a-
+--- +++
++-+- -+
+++--+ -
+-++--+Ø
ºŒ
ŒŒŒŒŒŒŒŒŒ
Œø
ßœ
œœœœœœœœœ
œ
, hence ttMM MM IoG
oG
oD
oDo ==8, and det detMMoG
oD()=()=+ + + + + + +()abgdezhq222222224.
Therefore, if o=1, MS OoG˛()8,R and MS OoD˛()8,R. Again, we only get a very tiny fraction
of SO 8,R() that way.
Also, and contrary to the case for the real numbers, the complex numbers and
the quaternions, in general MM MooG
oG
oG
¢¢„ and MM MooD
oD
oD
¢¢„, due to the non-associativity
of the product on O. For instance, ¢¢=-ie i, but MM M¢¢ -„iG
eG
iG .
If we try to do the same thing with hexadecimalions, we find that neither
    Mlh li j k e i j k e i j k e i j k i j k e i j k e i j k e i ja[] ¢ ¢ ¢ ¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢() ¢ ¢ ¢ ¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¢ ,, ,,,,,,, ,, , , , , , ,, ,,,,,,, ,, , , , , ,11 ¢¢¢ () ( )k nor its
right-hand version are rotation in general, even if l=1. That trail ends here as well!
10Chapter 2  Building the Quaternions
1- What to find here
This chapter, except for Section 5, only consists of well-known classical
results ([ N. Bourbaki (A) ], [S. Lang (1991) ],...). Some have been slightly restated (usually
with simplifications) from their original sources, but hardly anything new is presentedhere. In case the sources disagree on definitions, [
N. Bourbaki (A) ] will take precedence.
2- Cayley algebra, alternative algebra
Some of the structures we will be considering will not even be associative. To
save what may be, a weaker structure, which is interesting in its own right ispresented first. An algebra 
E is said to be alternative if the following trilinear
application, known as the associator of E, is alternating (which means its value is
zero if two of its arguments are identical):
  a:
,,EEE E·· ﬁ
() **()-*()* xyz x y z x y z a
This notion is interesting as, though an alternative algebra is not as wieldy as
an associative algebra, it is such that every sub-algebra engendered by any twoelements 
is associative. It also implies that an alternative algebra is a division
algebra (which means that for any x˛E, x„0, the applications   EEﬁ*;yx ya and
  EEﬁ*;yy xa are bijective, or that elements are “simplifiable”). In particular the
inverse of a non-zero element (if it exists) is unique in such an algebra.
The meat of this chapter is the following structure.Let 
A be a commutative ring, and E an algebra over A, not necessarily commutative
or associative, but having a unit element e (remember that since E is an A-algebra,
then "˛() "˛() ×= ×()*=* ×() ll l lAExx x x ee ).
A conjugation over E is any (there may be none) bijective, A-linear, function
s:EEﬁ such that:
1)see()=.
2)"()˛() *()=()*() xy x y y x,E2ss s  (beware the inversion of x and y!).
3)"˛() +()() ˛× xx x eEAs  and "˛() *()() ˛× xx x eEAs .
These properties imply6 "˛() *()=()* xx x x xEss , and7   "˛() ()= xx xEsso.
116xx e xx x xx x x xx x x xx x+()() ˛×Þ* ()=* + ()() -*= + ()() *-*= ()* ss s ssA .
7Given x˛E, there exists a˛A such that xx e+()=×sa ; the A-linearity of s then implies
  ss ss s a sxx x xe()+()=+()() =×() o , and finally, see()=. We will also write x for sx().If E is such an algebra, and if s is a conjugation over E, the structure
E,,, ,+*×()s is said to be a cayley algebra over A. On such a structure, it is convenient to
consider the cayley trace and cayley norm (an unfortunate misnomer as it is actuallyquadratic...), defined respectively by 
  TExx x()=+()s and   NExx x()=*()s.
Note that if E,+,*() has no zero divisors, for instance if it is a field, then
  NEx()=0 if and only if x=0.
We have the important relations:
  TTEEsxx()()=()
  NNEEsxx()()=()
  TTxy yx*()=*()
  TTT T T N N NEEE E E E E Exy yx x y x y x y x y*()() =* ()() =()*()-*()=+()-()-() ss
It is interesting to note that   TTxy yx*()=*() regardless of whether or not E is
associative or commutative. For the cayley norm, no such broad result seem to hold8;
however if E is alternative, then we also have   NN NEE Exy x y*()=()().
Finally, the following lemma will be useful for our purposes:
Lemma (Complexoïd): # Given x˛E, Vect ,Aex(), the A-module spanned by x and e, is
stable for *; it is a sub-cayley algebra of E which is both associative and commutative.
If xeˇ×A, let yex=× +×ab ,     MyGuy ue xe x=[] () () ()M a*,,,,  and     MyDuu ye xe x=[] () () ()M a*,,,, ;
then (with   T×()ex=TE and   N×()ex=NE)
MMMN
TyyG
yD===-
+Ø
ºŒø
ßœab
bb a
. Given ze x˛()Vect ,, we have MM M M M M**yz y z z y zy=== . $
# This is a simple consequence of the fact that  xx x e*=×-×TN , with   T×()ex=TE and
  N×()ex=NE! $
This lemma allows us, in particular, to define unambiguously the n-th power,
with n˛N, of any x˛E by the usual recursion rules, we will write the result, as usual
xn. It also trivially induces the following scholie:
Scholie (Powers): # Given x˛E, and n˛N, xe xn˛()Vect ,A and   NNEExxn n()=()(). $
3- The Cayley doubling procedure
It should be noted that this is simply the plain vanilla version of the doubling
process9; it will suffice here, however.
128Indeed, we have seen that such an equality does not hold for hexadecimalions!
9The general procedure involves abitrary coefficients which parameterize the operations.Let A be a commutative ring, and E,,, ,+*×()s a cayley algebra over A, not
necessarily commutative or associative, with unit element e. Let FEE=· and
eeF F=()˛,0; furthermore, let:
  +· ﬁ
()¢¢()() +¢+¢ ()FFF F:
,, , ,xy x y x x y y a
  *· ﬁ
()¢¢()() *¢-¢**¢+¢* ()FFF F:
,, , ,xy x y x x y yy x y x a
  ×· ﬁ
()() ××()FAF F:
,, ,ll lxy x y a
  s
sFFF:
,,ﬁ
() ()-() xy x y a
Proposition (Structure): # FFF F,,,+*×()  is an A-algebra, with unit eF, and sF is a conjugation
over F; F is associative if and only if E is both associative and commutative; F is
alternative if and only E is associative. Furthermore,   TTFExy x,()()=() and
  NN NFE Exy x y,()()=()+(). $
Keep in mind that since F is also an A-algebra then
"˛() "()˛() ×()=×()*()=()*×() ll l lAFFF F F F F F xy xy xy xy,, , , ee . It is interesting to note that, if
E is associative, we still have   NN NFF F Fxy x y xy x y,, , ,()*¢¢()() =()() ¢¢()(), even if F is not
associative.
Given the proposition, we can (and will) identify E with EE·{}0. Alternatively,
we can identify F with a superset of E. It is also possible to identify A with a subset
of E (and hence of F as well), in that case we have noted that all elements of A
commute with all elements of E, for the multiplication in  E, as well as with all
elements of F, for the multiplication in  F, even though E or F might not be commutative.
With this identification,   T and   N have value in A.
4- R, C, H, O, X...
We now consider AR= and ER=, with sxx()= and e=1, then   NRxx()=2 is
always positive (and zero if and only x=0, as R is a field). When we build F as above,
we get exactly C, and sF is the usual conjugation on C. We define i=()01;, and as
stated earlier, we identify R with R·{}0. As is well known, C is a commutative
field, in particular, real numbers commute with complex numbers. Due to ouridentifications, 
  TC and   NC have values in R, and actually, if zxy=+i then
  NNNC RRzxy x y z()=()+()=+=‡22 20, and   NCz()=0 if and only if z=0. We lose some of
the original properties of R as we build C, for instance we lose the existence of an
order compatible with the multiplication; we do get new and interesting properties at
13the same time, of course.Let’s do the doubling again, this time with AR= and EC=, with the usual
conjugation, and this time we get exactly H, the conjugation being the same as
defined earlier, given the definition of j=()01; and ki=()0;, and the identification of
C with C·{}0. Once again, we note that, as predicted, for quaternion multiplication,
real numbers commute with quaternions, though some quaternions do not commute(for instance 
ij ji*„*). As already stated H is a (non-commutative) field. Once again,
due to our new identifications,   TH and   NH have values in R, and actually,   NH is
always positive and   NHq()=0 if and only if q=0. We keep loosing original properties,
most notably the commutativity, when we go from C to H, but the new properties we
gain, notably the link with rotations in R3, which we saw earlier, still makes it
worthwhile. We also see that   THqq()=()2R e and   NHqqq()==22, as defined earlier.
There being not such thing as too much of a good thing, let’s do the doubling
once again, this time with AR= and EH=, and the conjugation just built on H. What
the process yields this time is known as the set of (Cayley) octonions, whose symbolis 
O. We, as is now usual, identify H with H·{}0. Yet again, we note that, for
octonion multiplication, real numbers commute with octonions, though some octonionsdo not commute (as some quaternions already do not commute). Yet again, due to ournew identifications, 
  TO and   NO have values in R, and actually,   NO is always positive
and   NOo()=0 if and only if o=0. The situation keeps deteriorating, though, as this
time the algebra is not associative anymore (but it is still associative). Octonions do
have uses, apart from being an example of a non-associative algebra. They can be usedto find a basis of non-vanishing vector field on 
S7 (the euclidian unit sphere in R8), in
the same way quaternions can be used to find one on S3, and complexes are used to
find one on S1. They also see use in theoretical physics ([ G. Dixon (1994) ]). Octonions
still are a division algebra, and non-zero octonions O have   NOOOO()[] ()-1s for inverse.
Despite the non-associativity of the multiplication, we still have   NN NOO Ooo o o*¢()=()¢(),
since the multiplication is associative on H.
We can keep doubling ad nauseam,  but things really get unwieldy. At the stage
after octonions, the hexadecimalions, X, the algebra is not even alternating. This
author does not know of any use the ulterior echelons may have been put to, if any.
5- The full Cayley ladder all at once
One might wonder if the whole doubling procedure might be “carried out to
infinity”. As it turns out, it can, after a fashion. We will present here a specialversion
10 of the global object, for simplicity.
Let A be a commutative ring, whose unit element will be called e.
Let us call   A0=A and s0 the identity over A. It is quite obvious that A,,,,+··() s
is a cayley algebra over A. Using the doubling procedure, we build   A1=·AA and s1,
and by induction we build   An and sn for all n˛N.
1410That is, the object built by our “plain vanilla” version of the Cayley doubling procedure.Consider AX[] the set of polynomials (in one indeterminate X) with coefficients
in A. We already have an A-algebra structure, which we will denote by AX[]+×·(),, , and
is the usual commutative algebra. We readily identify   A0=A with constant polynomials,
thru an homomorphism of A-modules   J0. It is trivial to see that   An identifies with
polynomials of degree less or equal to 21n-, thru the trivial A-modules isomorphism
  Jn. Let us call     IA Ann n xx :, ,ﬁ ()+1 0 a the canonical identification. Then     "˛() =+nnn nNJI J1o,
which means our identifications are all coherent.
So every element of every rung of the Cayley “ladder”, build by successively
doubling the preceding rung and begun by A, a finite number of times , can be identified
uniquely with some polynomial with coefficients in A, and conversely every element
of AX[] can be seen a some unique element of the Cayley ladder. As the multiplication
we will define differs, in general, from the polynomial multiplication, we will choosea new symbol for our construction.
Let 
  CA() be some set equipotent to AX[], the set of polynomials in one
indeterminate X over A, thru a bijection     IC:AA()[]aX. This bijection induces an
A-module on   CA(), from AX[]+×(),,, which we will denote by   CA()+×(),,, and we identify
  An with     I-1JAnn()().
We will now define a multiplication on   CA(), which we will denote by “ *”. Let
  pC˛()A and   qC˛()A; let   P=()Ip and   Q=()Iq, then there exists (at least) one n˛N such
that   Pnn˛()JA and   Qnn˛()JA. We chose the smallest such n. We now find the only   pnn˛A
such that   Ppnn=()J and the only   qnn˛A such that   Qqnn=()J. Finally     Ip q*()=*()JA nn npq
n.
We note that for all ¢>nn, we do have   Pnn˛()¢¢JA and   Qnn˛()¢¢JA and there are unique
  pnn¢¢˛A such that   Ppnn=()¢¢J and   qnn¢¢˛A such that   Qqnn=()¢¢J, but thanks to the coherence
of the identifications we also have   JJAA nn n nn n pq p q
n n*() =*()¢¢ ¢¢.
It is easy to verify that   CA()+×*(),, , is an A-algebra. However, in general
  Ip q Ip Iq*()„()·(). For instance if AR= then   Xi1=()I,   Xj2=()I and   Xk3=()I, and thus
  II IiXXXX j k()=„=·= ()·()1523. So   I is not, in general, an algebra isomorphism between
  CA()+×*(),, , and AX[]+×·(),, ,, as stated earlier.
We likewise define the conjugation s, and the cayley trace and “norm”, over
  CA() thru the identifications   Jn, with values in   A0. It is now easy to check that
  CA()+×*(),, ,,s is a cayley algebra over A (usually not commutative or associative),
which, thru the identifications, contains all the rungs of the cayley doubling procedure
starting with A. Elements of A commute with all elements of   CA(), for *.
We will shortly use the fact that if       Ip()=+ + + = ()--aa a01 2121XX pnn
nn L J, then
      Ip p*()=*() =-+ +()() ++ +---JA nn npp X X
nnnnaa a a a a a02
12
212
10 2102122 LL ; this is simply proved
by recurrence.
As a first example of Cayley ladders, let us consider   CZZ2()+·*(),,,, I d. It is a
commutative and associative cayley algebra, the conjugation being the identity on
  CZZ2(); however it has zero divisors, as for instance   II--+()*+()=1111 0XX , but if
15  pC˛()ZZ2 and   Ip()has an odd number of 1 then   Ip p*()=1.The second, perhaps more interesting example, is   CR(), which we have actually
used already. In that case we can see that if   aC˛()R, then     NCaR()() is always positive,
and     NCaR()()=0 if and only if   a=0; furthermore,
    aa a a a a aCC „Þ ()() ()Ø
ºŒø
ßœ*=* ()() ()Ø
ºŒø
ßœ=()-
()-0111NNRRss . It is also possible in this case to compute
square roots! Indeed, let   xn˛A, with     JnxAA X AXnn()=+ ++--
01 2121L ; we seek   yn˛A with
    JnyX Xnn()=+ + +--aa a01 2121L  such that yy x*=. This amounts to solving, in R2n
 the
system:
  aa a
aa
aa02
12
212
0
10 1
210212
2-+() =
=
=ì
íï
ï
îï
ï-
--L
Mn
nnA
A
A
This system is easily solved by considering first the case x=0, for which there is a
unique solution y=0, second the subcase x˛+R* for which there are exactly two
solutions given by yx=–, third the subcase xˇR (if n‡1, of course) for which there
are also exactly two solution given by 
    JnyAXAXn n()=+ + +- -aaa01
021
021
22L  with
a02
2=– ()+Rexx, and finally the case x˛-R*, for which the solutions are all the
  yn˛A such that   Ry()=0 and yx=.
This means that the solutions to yx2=, where xˇ-R are the same in every rung
of the real Cayley ladder (that is, there are exactly two, opposite solutions, belongingto the same rung), and the solution to 
y20= is always y=0, in whatever rung of the
Cayley ladder. However, solutions to yx2= for x˛-R* differ depending upon the precise
rung: in R there is no solution, in C there are exactly two, opposite, solutions, in H
and above there is an innumerable number of solutions (full spheres)!
Note that in any case a y such that yx2= commutes with x, but that two such
solutions need not commute with each other!
At least two topologies are interesting to consider on   CR(): the norm topology
induced by the square root of   N, the Cayley “norm” on   CR() (we will write   cc=()N),
which we will call   T, and the strict inductive limit topology ([ V.-K. Khoan (1972)])
defined by the rungs     Cnn n=J A() of   CR() on which we consider the norms qnn=×-()202sup ,,
which we will call   T¥.
The problem with × is that it is not an algebra norm, as evidenced by the
hexadecimalions. Furthermore,     CR()(),T is not complete, its completion being   l2R()
with its usual topology.
On the other hand,     CR()()¥,T is complete, and the product is (trivially) separately
16continuous ([ N. Bourbaki (EVT)]), but it is not known if it is continuous.For both topologies, any finite-dimensional vector space is closed and the
restriction to that vector space is just the usual (euclidian) topology.
At any rate, given   x˛()CR, the Powers Scholie proves that Vect , ,R1x()()  is a
commutative R-Banach algebra (of dimension 1 if and only if x˛R).
As a final thought, since   CA() is an A-Cayley algebra, we can perform the
Cayley doubling procedure on it! And again, and so on and so forth... We can actuallyperform an infinity of doubling as above, and embed all these doublings in what,essentially, is 
AXY,[]. And then we can start all over again... As we can readily see,
there is no “ultimate” step... What seems to be going on is that we can build an objectfor any 
finite ordinal ([ J.-M. Exbrayat, P. Mazet (1971)]), and we have built an object,
which we have called   CA(), for the first infinite ordinal w. We have then seen that the
doubling of   CA() yields the object corresponding to w* (the successor of w). The next
infinite ordinal with no predecessor ( 2w) corresponds to AXY,[]. Further on
(corresponding to w2), we find the set of polynomials in an indeterminate number of
indeterminates ( i.e.11 ANN[][]). It is not clear, however, in which way we can extend the
construction to any set of ordinals ( i.e. there is no clear transfinite “recurrence
formula”).
1711Recal that if X is a monoïd ( [N. Bourbaki (A)] ) and Y is a set, XY[] is the set of functions from Y to X
which take values different from the neutral ement of X only for a finite numbers of elements of Y.Chapter 3  The Exponential
1- What to find here
This chapter is mostly designed to prove the explicit formula for the exponential
in   CR(), and give several related results. As far as I known, these results are new.
There are many notions of the exponential, and many ways to see several of
them. These, of course, agree when various different definitions can be put forwardfor the same object to be exponentiated. We will be concerned here mainly with theanalyst’s point of view, and define the exponential of quaternions thru the use of theusual power series ([
A.F. Beardon (1979)],...). It is known that the approach detailed in
[F. Pham (1996)] could also be used, at least for quaternions, though I believe it would
then be necessary to derive the power series representation (or the intermediarydifferential representation we will also use) to achieve our present goal. It remainsto be seen if it can also be carried over to the whole of 
  CR().
2- Definition
Given   x˛()CR, we will call exponential of x, and we will write Expx() the
element of   CR() given by x
nn
n!=+¥
å
0. The unambiguity and existence of Expx() is given by the
fact that Vect , ,R1x()()  is a commutative R-Banach algebra, as we have said earlier.
This, of course agrees with the definition on R and C. We must bear in mind that
Exp Vect ,xx()˛()R1.
We see at once that   "˛()() ()=() xx xCRExp Exp . The exponential is continuous when
restricted to each rung of   CR(), and has its values into the same rung (we will give a
more precise result later on).
3- Links with differentiation
Differentiating a function of one or several quaternions (or higher in the Cayley
ladder) is quite problematic. Of course, since Vect ,R1x() is commutative, there is no
ambiguity in defining ff /yx y x()-()() -() if yx˛()Vect ,R1, and we can therefore differentiate
ExpVect ,R1x() with respect to some yx˛()Vect ,R1 and find that it is once again ExpVect ,R1x().
It is more fruitful, however, to differentiate a function of a real variable, with
values in some topological R-vector space.
Let us therefore consider, for some   x˛()CR, the function     et t xx: , ExpRRﬁ() ()[] C a .
It is clear that ex takes its values in Vect ,R1x(), is differentiable and et x e t e t xxx x¢()=()=(),
and of course ex01()=. This, of course proves that ex is the unique solution to ¢=fx f,
f01()= in         C11 RR,Vect , x()() , the set of one-time continuously differentiable functions
from R to Vect ,R1x(). Given any rung   E of   CR() such that   x˛E, ex is still the unique
solution to ¢=fx f, f01()= in           C1R,E().
The perhaps surprising phenomenon is when we consider the equation ¢=fx f,
f0()=g in           C1R,E() for some rung   E of   CR() such that   x˛E, and   g˛E. If   E=R or   E=C,
18then of course the solution is etx()g, and it turns out this is still true if   E=H,because of the associativity of the quaternionic product (this, actually, is how one
can navigate the unit sphere of the quaternions, which is useful for interpolatingbetween orientations, and was the problem under examination in [
D. Pletincks (1989) ]). It
is interesting to note that this is still true if   E=O because of the alternative nature
of that algebra. This stops to be true with hexadecimalions, however. Indeed, consider
xie=+¢¢¢ and g=j, and let gte tx()=()g. We will shortly see that ei eie+¢¢¢æ
Łçö
ł÷=+ ¢¢¢()p2
42
2,
from which we can deduce gp2
42
2æ
Łçö
ł÷=+ ¢¢¢()iej and ¢æ
Łçö
ł÷=+¢¢¢() +¢¢¢()æ
Łçö
ł÷=- gp2
42
22 ie ie j j
whereas ie ie iej jk+¢¢¢()æ
Łö
ł=+¢¢¢() +¢¢¢()æ
Łçö
ł÷=- + ¢¢¢() gp
22
22 , and therefore ¢æ
Łö
ł„+¢¢¢()æ
Łö
łggpp
22ie .
Numerical integration procedures will yield the solution to the differential equation,
and therefore not the exponential function, unless care has been taken to chose thestarting point correctly.
4- The closed formula for the exponential in   CR()
We now give the main result of this work. Note that it is closed only in that we
assume the exponential and classical trigonometric functions on R to be givens12.
Theorem (Exponential):# If   x˛()CR then Exp e cos Ur sinc Ur UrRexx x xx()= ()()+ ()() () []()
p . $
# Let   y˛()CR such that Rey()=0, y=1; then   yy y yy21 =()-[] =-()=- TN . Therefore, in
Vect ,R1y() computations are carried out exactly as in C, with y taking the place of i.
More precisely,   C, +Rﬁ() + []Vect ,1y a ib a by a is a Banach isomorphism.
Let now   x˛()CR. If x˛R, we see the result is trivially true. Assume, then that xˇR,
and let ÃUr
Urxx
x=()
(). Then ReÃx()=0, Ãx=1, xx x x=()+() Re Ur Ã, and of course
Vect , Vect , ÃRR11xx()=(). The previous identification then allows us to find
Exp e cos Ur sin Ur ÃRexx x xx()= ()()+()() [](). $
As an example, we have, as announced earlier, Expp2
42
2ie ie+¢¢¢()æ
Łçö
ł÷=+ ¢¢¢().
1912A family of special functions will be of interest here, that of the “Sinus Cardinal” functions, defined for
some parameter a˛R+* by 
  sinc : ,sin
a xx
a
x
aRRﬁæ
Łö
łØ
ºŒ
ŒŒ
Œø
ßœ
œœ
œap
. We will, by similitude, define the
“Hyperbolic Sinus Cardinal” family of functions defined for some parameter a˛R+* by
  sinhc : ,sinh
a xx
a
x
aRRﬁæ
Łö
łØ
ºŒ
ŒŒ
Œø
ßœ
œœ
œap
. These functions are entire functions on all of R.5- Some properties of the exponential and further consequences
We compute at once Exp eRe xx()=().
As should be expected when we lose the benefit of commutativity, the exponential
of a sum is in general different from the product of the exponentials; for instance wehave 
Exp Exp cos cos sin cos sin cos sin sinij i j k()()=() ()+() ()+() ()+() () 11 11 11 11  whereas
Exp cosij ij+()=()++()22
2. We also see immediately that the exponential is not injective
on any rung   E of   CR() containing C, as it is already not injective on C! We, however
also lose the periodicity when   E contains H, as the periods would make an additive
subgroup of   E but the solutions of Exp x()=1 on H are exactly the set 202p..NS{}· (with
S2 the unit sphere of R3); the rest is number theory (and trying to fit square pegs into
round holes). We have more details on the surjectivity of the exponential:
Corollary (sujectivity): # If   E is a rung of   CR() containing C, then the exponential is a
surjection from   E onto   E-{}0. $
# We first note that given any   x˛()CR, Exp eRe xx()=() proves that the exponential never
take the value 0 on   CR().
Let now   y˛E, y„0. If y˛R we know we can solve our problem (in R if y>0, in C if
y<0). Assume therefore that yˇR. We can find r˛R such that er=y. Let Äyy
y=;
Äy=1 and ÄyˇR, so let ÃUrÄ
UrÄyy
y=()
(), so that ÄReÄUrÄÃ yy y y=()+(), ReÄy()„0 and ReÄUrÄ yy()+()=2 21.
Let qp˛][0; the unique number such that cos Re Äq()=()y and sin Ur Ä q()=()y. We see that
Exp Ãrq+()=yy. $
We can likewise find closed formulæ for other interesting entire functions
(defining cos!xx
nnn
n()=-()
()=+¥
å1
22
0, sin!xx
nnn
n()=-()
+()+
=+¥
å1
2121
0, cosh!xx
nn
n()=()=+¥
å2
02, sinh!xx
nn
n()=+()+
=+¥
å21
021), to wit:
cos cos Re cosh Ur sin Re sinhc Ur x Ur xxxx x()=()() ()()-()() ()() ()p
sin sin Re cosh Ur x cos Re sinhc Ur x Ur xxx x()=()() ()()+()() ()() ()p
cosh cosh Re cos Ur sinh Re sinc Urxx x x x()= ()() ()()+()() ()()p
sinh sinh Re cos Ur cosh Re sinc Ur Urxx x xx x()=()() ()()+ ()() ()() ()p
and of course many other such.
6- Conclusion
We have found a closed formula for the exponential, for quaternions, octonions,
and beyond.
An interesting application of this formula is navigation on the unit sphere of
the quaternions, leading to an algorithm for the interpolation of orientations, butwhich, in general, does not preserve the horizontal. This can also be achieved, however,and has been implemented by the author and a colleague ([
Horizontal-preserving quaternions ]).
20Bibliography
A.F. Beardon (1979): Complex Analysis, The Argument Principle in Analysis and Topology; John Wiley &
Sons, A Wiley-Interscience Publication, 1979.
M. Berger (1990): Géom etrie 1; Nathan, 1990.
N. Bourbaki (A): Algèbre.
N. Bourbaki (EVT): Espaces Vectoriels Topologiques.
P. de Casteljau (1987): Les quaternions; Hermes, Traité des nouvelles technologies, série mathématiques
appliquées, 1987.
G. Dixon (1994): D ivision Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design
of Physics; Kluwer Academic Publishers, Mathematics and Its Applications, 1994.
J.-M. Exbrayat, P. M azet (1971): al gèbre 1, Notions Fondamentales de la Théorie des Ensembles; Hatier-
Université, Notions Modernes de Mathématiques, 1971.
K. Gürlebeck, W. Spössig (1989): Quaternion Analysis and Elliptical Boundary Problems; Birkhaüser,
International Series of Numerical Mathematics, vol. 89, 1989.
T. Howell,  J.C. Lafon (1975): The complexity of the quaternion product; Cornell Computer Science TR
75-245, June 1975.
V.-K. Khoan (1972): D istributions Analyse de Fourier Opérateurs aux Dérivées Partielles, tome 1, Cours
et exercices résolus maitrise de mathématiques certificat C2; Vuibert, 1972.
S. Lang (1971): Algebra; Addison-Wesley Series in Mathematics, Addisson-Wesley Publishing
Company, Revised printing: January 1971.
D. Leborgne (1982): Calcul différentiel et géometrie; P.U.F., Mathématiques, 1982.
J. Lelong-Ferrand, J.M. Arnaudiès (1978): Co urs de mathèmatiques, Tome 1, Algèbre (3eme édition);
Dunod Université, 1978.
C.W. Misner, K.S. Thone, J.A. Wheeler (1973): Gravitation; W.H. Freeman and Company, New York,
Chapter 41 -Spinors,1973.
F. Pham (1996): Une définition non standard de l’exponentielle (variation sur un thème de Pierre
Cartier), 1996 (early version, unpublished, presented at the Journées Non Standard,Paris, 14/12/1996 and perhaps elsewhere).
G. Pichon (1973): Groupes de Lie, representations linéaires et applications; Hermann, Collection
Méthodes, Paris, 1973.
D. Pletincks (1989): Quaternion calculus as a basic tool in computer graphics; 
in “The Visual Computer,
an International Journal of Computer Graphics” vol. 5, n° 1/2, pp 2-13, 1989.
S. Salamon (1982): Quaternionic Kaeler Manifolds; in Inventiones Mathematica, vol. 67, n° 1,
pp 143-171.
A. Sudbery (1979): Quaternionic Analysis; in Proceedings of the Cambridge Philosophical Society,
vol. 85, 1979.
Software index
Horizontal-preserving quaternions: available for licencing from the author, © Hubert Holin & Didier Vidal.
Maple : A commercial computer-aided mathematics software, currently in version V, release
5.1; edited by Waterloo Maple Inc., 450Phillip St., Waterloo, ON N2L SJ2, C anada;
http://www.maplesoft.com.
Interesting URLs
G. Dixon: http://www.7stones.com/Homepage/sevenhome2.html.
21E. Weisstein: http://www.treasure-troves.com/math/CayleyNumber.html.