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.

已经考虑了以\(8 \times 8\)矩阵表示的扩展伽马矩阵 Clifford–Dirac 和 SO(1,9) 代数。提出了 8 分量狄拉克方程的 Clifford 代数的 256 维伽马矩阵表示。两个同构实现\(\textit{C}\ell ^{\texttt {R}}\) (0,8) 和\(\textit{C}\ell ^{\texttt {R}}\) (1, 7) 考虑。还介绍了 45 维 SO(10) 和 SO(1,9) 代数的相应伽玛矩阵表示，其中包含标准和附加自旋算子。 SO(10), SO(1,9) 和相应的\(\textit{C}\ell ^{\texttt {R}}\) (0,8) \(, \textit{C}\ell ^ {\texttt {R}}\) (1,7) 表示被确定为实数域上的代数。建议的李代数 SO(10)、SO(1,9) 的伽马矩阵表示是基于 Clifford 代数\(\textit{C}\ell ^{\texttt {R}}\) (0 ,8) \(, \textit{C}\ell ^{\texttt {R}}\) (1,7) 表示。演示了与标准 4 分量狄拉克旋量空间中相应代数的比较。所提出的数学对象允许推广我们的结果，这些结果是先前为标准狄拉克方程、更高自旋方程以及特别是描述自旋为 3/2 的粒子的方程而获得的。找到了 Foldy-Wouthuysen 表示中 8 分量狄拉克方程的最大不变性纯矩阵代数 84 维。狄拉克方程在普通表示中的相应对称性也被发现。简要讨论了所考虑的李代数对任意维 SO(n) 和 SO(m,n) 的可能推广。