  This paper presents some new Lie groups preserving fixed subspaces of
geometric algebras (or Clifford algebras) under the twisted adjoint
representation. We consider the cases of subspaces of fixed grades and
subspaces determined by the grade involution and the reversion. Some of the
considered Lie groups can be interpreted as generalizations of Lipschitz groups
and spin groups. The Lipschitz groups and the spin groups are subgroups of
these Lie groups and coincide with them in the cases of small dimensions. We
study the corresponding Lie algebras.
