# An Algorithm for Fast Multiplication of Kaluza Numbers
## 2 Preliminary Remarks
\begin{bmatrix}d_{10}&d_{11}&d_{8}&d_{9}&d_{14}&d_{15}&d_ {12}&d_{13}\\ -d_{11}&d_{10}&d_{9}&d_{8}&d_{15}&d_{14}&d_{13}&d_{12}\\ -d_{2}&d_{3}&d_{0}&d_{1}&d_{6}&d_{7}&d_{4}&d_{5}\\ -d_{3}&d_{2}&d_{1}&d_{0}&d_{7}&d_{6}&d_{5}&d_{4}\\ -d_{26}&d_{27}&d_{24}&d_{25}&d_{30}&d_{31}&d_{28}&d_{29}\\ d_{27}&d_{26}&d_{25}&d_{24}&d_{31}&d_{30}&d_{29}&d_{28}\\ d_{18}&d_{19}&d_{16}&d_{17}&d_{22}&d_{23}&d_{20}&d_{21}\\ d_{19}&d_{18}&d_{17}&d_{16}&d_{23}&d_{22}&d_{21}&d_{20}\end{bmatrix} ,\]\[\mathbf{B}_{8}^{(1+)}= \begin{bmatrix}d_{30}&d_{31}&d_{28}&d_{29}&d_{26}&d_{27}&d_{24 }&d_{25}\\ d_{31}&d_{30}&d_{29}&d_{28}&d_{27}&d_{26}&d_{25}&d_{24}\\ -d_{22}&d_{23}&d_{20}&d_{21}&d_{18}&d_{19}&d_{16}&d_{17}\\ d_{23}&d_{22}&d_{21}&d_{20}&d_{19}&d_{18}&d_{17}&d_{16}\\ d_{14}&d_{15}&d_{12}&d_{13}&d_{10}&d_{11}&d_{8}&d_{9}\\ d_{15}&d_{14}&d_{13}&d_{12}&d_{11}&d_{10}&d_{9}&d_{8}\\ d_{6}&d_{7}&d_{4}&d_{5}&d_{2}&d_{3}&d_{0}&d_{1}\\ -d_{7}&d_{6}&d_{5}&d_{4}&d_{3}&d_{2}&d_{1}&d_{0}\end{bmatrix} ,\]  
\[\mathbf{B}_{8}^{(1-)}= \begin{bmatrix}d_{20}&d_{21}&d_{22}&d_{23}&d_{16}&d_{17} &d_{18}&d_{19}\\ d_{21}&d_{20}&d_{23}&d_{22}&d_{17}&d_{16}&d_{19}&d_{18}\\ -d_{28}&d_{29}&d_{30}&d_{31}&d_{24}&d_{25}&d_{26}&d_{27}\\ d_{29}&d_{28}&d_{31}&d_{30}&d_{25}&d_{24}&d_{27}&d_{26}\\ d_{4}&d_{5}&d_{6}&d_{7}&d_{0}&d_{1}&d_{2}&d_{3}\\ d_{5}&d_{4}&d_{7}&d_{6}&d_{1}&d_{0}&d_{3}&d_{2}\\ d_{12}&d_{13}&d_{14}&d_{15}&d_{8}&d_{9}&d_{10}&d_{11}\\ d_{13}&d_{12}&d_{15}&d_{14}&d_{9}&d_{8}&d_{11}&d_{10}\end{bmatrix} .\]  
In order to simplify, we introduce the following notation for the elements of matrix \(\mathbf{B}_{8}^{(2+)}\) (14):  
\[ \begin{array}{llllll}c_{32}=b_{12}+b_{15},&c_{33}=b_{18}+b_{21},&c_{34}=b5+b _{24},&c_{35}=b9+b_{28},\\ c_{36}=b_{14}-b_{30},&c_{37}=b_{20}-b_{31},&c_{38}=b_{23}-b_{25},&c_{39}=b_{2 9}-b_{27},\\ c_{40}=b_{24}-b5,&c_{41}=b_{28}-b9,&c_{42}=b_{12}-b_{15},&c_{43}=b_{21}-b_{18},\\ c_{44}=b_{23}+b_{25},&c_{45}=b_{27}+b_{29},&c_{46}=b_{14}+b_{30},&c_{47}=b_{20 }+b_{31},\end{array} \]  
we obtain:  
\[\mathbf{B}_{8}^{(2+)}= \begin{bmatrix}-c_{32}&c_{33}&c_{34}&c_{35}&c_{36}&c_{ 37}&c_{38}&c_{39}\\ -c_{33}&c_{32}&c_{35}&c_{34}&c_{37}&c_{36}&c_{39}&c_{38}\\ c_{40}&c_{41}&c_{42}&c_{43}&c_{44}&c_{45}&c_{46}&c_{47}\\ c_{41}&c_{40}&c_{43}&c_{42}&c_{45}&c_{44}&c_{47}&c_{46}\\ -c_{46}&c_{47}&c_{44}&c_{45}&c_{42}&c_{43}&c_{40}&c_{41}\\ -c_{47}&c_{46}&c_{45}&c_{44}&c_{43}&c_{42}&c_{41}&c_{40}\\ c_{38}&c_{39}&c_{36}&c_{37}&c_{34}&c_{35}&c_{32}&c_{33}\\ -c_{39}&c_{38}&c_{37}&c_{36}&c_{35}&c_{34}&c_{33}&c_{32}\end{bmatrix} .\]  
Now, we introduce the following notation for the elements of matrix \(\mathbf{B}_{8}^{(2-)}\) (15):  
\[ \begin{array}{llllll}c_{48}=b_{12}-b_{15},&c_{49}=b_{18}-b_{21},&c_{50}=b5-b_ {24},&c_{51}=b_{28}-b9,\\ c_{52}=b_{14}+b_{30},&c_{53}=b_{20}+b_{31},&c_{54}=b_{23}+b_{25},&c_{55}=b_{27}+ b_{29},\\ c_{56}=b5+b_{24},&c_{57}=b9+b_{28},&c_{58}=b_{12}+b_{15},&c_{59}=b_{18}+b_{21}, \\ c_{60}=b_{23}-b_{25},&c_{61}=b_{27}-b_{29},&c_{62}=b_{30}-b_{14},&c_{63}=b_{20 }-b_{31},\end{array} \]  
we obtain:  
\[\mathbf{B}_{8}^{(2-)}= \begin{bmatrix}c_{48}&c_{49}&c_{50}&c_{51}&c_{52}&c_{53}& c_{54}&c_{55}\\ c_{49}&c_{48}&c_{51}&c_{50}&c_{53}&c_{52}&c_{55}&c_{54}\\ c_{56}&c_{57}&c_{58}&c_{59}&c_{60}&c_{61}&c_{62}&c_{63}\\ c_{57}&c_{56}&c_{59}&c_{58}&c_{61}&c_{60}&c_{63}&c_{62}\\ -c_{62}&c_{63}&c_{60}&c_{61}&c_{58}&c_{59}&c_{56}&c_{57}\\ c_{63}&c_{62}&c_{61}&c_{60}&c_{59}&c_{58}&c_{57}&c_{56}\\ -c_{54}&c_{55}&c_{52}&c_{53}&c_{50}&c_{51}&c_{48}&c_{49}\\ -c_{55}&c_{54}&c_{53}&c_{52}&c_{51}&c_{50}&c_{49}&c_{48}.\end{bmatrix} .\]All of the above matrices have the same internal structure. We can permute rows and columns using the \(\pi_{r}=\) (5 1 2 7 4 0 3 6) and \(\pi_{c}=\) (5 1 2 6 4 0 3 7) permutation rules, respectively.