  Extended gamma matrix Clifford--Dirac and SO(1,9) algebras in the terms of $8
\times 8$ matrices have been considered. The 256-dimensional gamma matrix
representation of Clifford algebra for 8-component Dirac equation is suggested.
Two isomorphic realizations $\textit{C}\ell^{\texttt{R}}$(0,8) and
$\textit{C}\ell^{\texttt{R}}$(1,7) are considered. The corresponding gamma
matrix representations of 45-dimensional SO(10) and SO(1,9) algebras, which
contain standard and additional spin operators, are introduced as well. The
SO(10), SO(1,9) and the corresponding $\textit{C}\ell^{\texttt{R}}$(0,8)$,
\textit{C}\ell^{\texttt{R}}$(1,7) representations are determined as algebras
over the field of real numbers. The suggested gamma matrix representations of
the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford
algebras $\textit{C}\ell^{\texttt{R}}$(0,8)$, \textit{C}\ell^{\texttt{R}}$(1,7)
representations. Comparison with the corresponded algebras in the space of
standard 4-component Dirac spinors is demonstrated. The proposed mathematical
objects allow generalization of our results, obtained earlier for the standard
Dirac equation, for equations of higher spin and, especially, for equations,
describing particles with spin 3/2. The maximal 84-dimensional pure matrix
algebra of invariance of the 8-component Dirac equation in the
Foldy--Wouthuysen representation is found. The corresponding symmetry of the
Dirac equation in ordinary representation is found as well. The possible
generalizations of considered Lie algebras to the arbitrary dimensional SO(n)
and SO(m,n) are discussed briefly.
