  In this note, we use give some algebraic applications of a previous result by
the author which compares the deformations parameterized by the Maurer-Cartan
elements of a differential graded Lie algebra, and a differential graded Lie
subalgebra: It gives a criterion for the map on the space of Maurer-Cartan
elements up to gauge equivalence, induced by the inclusion of the subalgebra,
to be locally surjective. By making appropriate choices for the differential
graded Lie algebra and a differential graded Lie subalgebra, we recover some
classical results in the deformation theory of finite-dimensional Lie and
associative algebras. We consider rigidity of Lie and associative algebras, Lie
algebra morphisms, and give a quick proof of the fact that the deformation
theories of a unital associative algebra as a unital associative algebra and as
an associative algebra are equivalent. We then turn to stability of Lie
subalgebras and their morphisms under deformations of the ambient Lie algebra
structure, and study when the compatibility with a geometric structure of a
representation of a Lie algebra on a vector space is stable under deformations
of the representation.
