  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.
