  We show that the Tits index $E_8^{133}$ cannot be obtained by means of the
Tits construction over a field with no odd degree extensions. We construct two
cohomological invariants, in degrees 6 and 8, of the Tits construction and the
more symmetric Allison-Faulkner construction of Lie algebras of type $E_8$ and
show that these invariants can be used to detect the isotropy rank.
