# An Algorithm for Fast Multiplication of Kaluza Numbers
## 2 Preliminary Remarks
We obtain the following form:  
\[\mathbf{B}_{8}^{(\gamma)}=\mathbf{P}_{8}^{(r)}\mathbf{\hat{B}}_{8}^{(x)} \mathbf{P}_{8}^{(c)}, \tag{16}\]  
where \(\mathbf{B}_{8}^{(\gamma)}\), \(\mathbf{\hat{B}}_{8}^{(\gamma)}\) are the corresponding items in the sets:  
\[\mathbf{B}_{8}^{(\gamma)}\in\Big{\{}\mathbf{B}_{8}^{(0+)},\, \mathbf{B}_{8}^{(0-)},\,\mathbf{B}_{8}^{(1+)},\,\mathbf{B}_{8}^{(1-)},\, \mathbf{B}_{8}^{(2+)},\,\mathbf{B}_{8}^{(2-)}\Big{\}},\] \[\mathbf{\hat{B}}_{8}^{(\gamma)}\in\Big{\{}\mathbf{\hat{B}}_{8}^{ (0+)},\,\mathbf{\hat{B}}_{8}^{(0-)},\,\mathbf{\hat{B}}_{8}^{(1+)},\,\mathbf{ \hat{B}}_{8}^{(1-)},\,\mathbf{\hat{B}}_{8}^{(2+)},\,\mathbf{\hat{B}}_{8}^{(2- )}\Big{\}}\]  
and  
\[\mathbf{P}_{8}^{(r)}= \begin{bmatrix}0&0&0&0&0&1&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&1&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&1&0&0&0&0\end{bmatrix} ,\quad\quad\mathbf{P}_{8}^{(c)}= \begin{bmatrix}0&0&0&0& 0&1&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&1&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&0&1\end{bmatrix} .\]  
The matrices \(\mathbf{\hat{B}}_{8}^{(\gamma)}\) (16) are calculated via the following equation:  
\[\mathbf{\hat{B}}_{8}^{(\gamma)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1} \mathbf{B}_{8}^{(x)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}\]  
and have a standardized form (9) that reduces the number of multiplications. Thus, we can write:  
\[\mathbf{\hat{B}}_{8}^{(0+)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(0+)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(0)}&\mathbf{B}_{4}^{(1)}\\ \mathbf{B}_{4}^{(1)}&\mathbf{B}_{4}^{(0)}\end{bmatrix} ,\]  
\[\mathbf{\hat{B}}_{8}^{(0-)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(0-)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(2)}&\mathbf{B}_{4}^{(3)}\\ \mathbf{B}_{4}^{(3)}&\mathbf{B}_{4}^{(2)}\end{bmatrix} ,\]  
\[\mathbf{\hat{B}}_{8}^{(1+)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(1+)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(4)}&\mathbf{B}_{4}^{(5)}\\ \mathbf{B}_{4}^{(5)}&\mathbf{B}_{4}^{(4)}\end{bmatrix} ,\]  
\[\mathbf{\hat{B}}_{8}^{(1-)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(1-)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(6)}&\mathbf{B}_{4}^{(7)}\\ \mathbf{B}_{4}^{(7)}&\mathbf{B}_{4}^{(6)}\end{bmatrix} ,\]  
\[\mathbf{\hat{B}}_{8}^{(2+)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(2+)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(8)}&\mathbf{B}_{4}^{(9)}\\ \mathbf{B}_{4}^{(9)}&\mathbf{B}_{4}^{(8)}\end{bmatrix} ,\]  
\[\mathbf{\hat{B}}_{8}^{(2-)}=\Big{(}\mathbf{P}_{8}^{(r)}\Big{)}^{-1}\mathbf{B} _{8}^{(2-)}\Big{(}\mathbf{P}_{8}^{(c)}\Big{)}^{-1}= \begin{bmatrix}\mathbf{B}_{ 4}^{(10)}&\mathbf{B}_{4}^{(11)}\\ \mathbf{B}_{4}^{(11)}&\mathbf{B}_{4}^{(10)}\end{bmatrix} ,\]  
where:  
\[\mathbf{B}_{4}^{(0)}= \begin{bmatrix}d_{20}&d_{16}&d_{19}&d_{23}\\ d_{4}&d_{0}&d_{3}&d_{7}\\ d_{13}&d_{9}&d_{10}&d_{14}\\ -d_{28}&d_{24}&d_{27}&d_{31}\end{bmatrix} ,\quad\mathbf{B}_{4}^{(1)}= \begin{bmatrix} d_{21}&d_{17}&d_{18}&d_{22}\\ d_{5}&d_{1}&d_{2}&d_{6}\\ d_{12}&d_{8}&d_{11}&d_{15}\\ d_{29}&d_{25}&d_{26}&d_{30}\end{bmatrix} ,\]\[\mathbf{B}_{4}^{(2)}= \begin{bmatrix}d_{30}&d_{26}&d_{25}&d_{29}\\ d_{14}&d_{10}&d_{9}&d_{13}\\ -d_{7}&d_{3}&d_{0}&d_{4}\\ -d_{22}&d_{18}&d_{17}&d_{21}\end{bmatrix} ,\quad\mathbf{B}_{4}^{(3)}= \begin{bmatrix} d_{31}&d_{27}&d_{24}&d_{28}\\ d_{15}&d_{11}&d_{8}&d_{12}\\ d_{6}&d_{2}&d_{1}&d_{5}\\ d_{23}&d_{19}&d_{16}&d_{20}\end{bmatrix} ,\]  
\[\mathbf{B}_{4}^{(4)}= \begin{bmatrix}d_{10}&d_{14}&d_{13}&d_{9}\\ -d_{26}&d_{30}&d_{29}&d_{25}\\ d_{19}&d_{23}&d_{20}&d_{16}\\ -d_{2}&d_{6}&d_{5}&d_{1}\end{bmatrix} ,\quad\mathbf{B}_{4}^{(5)}= \begin{bmatrix} -d_{11}&d_{15}&d_{12}&d_{8}\\ d_{27}&d_{31}&d_{28}&d_{24}\\ d_{18}&d_{22}&d_{21}&d_{17}\\ -d_{3}&d_{7}&d_{4}&d_{0}\end{bmatrix} ,\]  
\[\mathbf{B}_{4}^{(6)}=