On inner automorphisms preserving subspaces of Clifford algebras

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras – subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras – graded central simple algebras or graded central simple algebras with involution.