) 0. Introduction. Let L = (V, ¡j) be a Lie (resp. associative) algebra with underlying vector space V and multiplication et. A subalgebra S of L is stable if 5 remains a subalgebra under small deformations of L. That is, if L'=(V, p.') is a Lie (resp. associative) algebra with p.' near c¿, there exists a subalgebra S' of L' which is isomorphic to S and whose underlying vector space is near that of S. (See §4 for a precise definition.) In a letter to one of the authors, J. M. G. Fell has conjectured that if 5 is a semisimple subalgebra of a finite-dimensional real Lie algebra, then S is stable. The following "stability theorem" shows that this is indeed the case.Since submitting this paper, we have learned that several of our results for associative algebras, in particular Corollary 11.4(h), were obtained independently by F. J. Flanigan in his 1966 Dissertation at the University of California, Berkeley.Theorem. Let L=(V,p.) be a finite-dimensional Lie (resp. associative) algebra over either an algebraically closed field or the field R of real numbers. Let S be a subalgebra ofL such that H2(S, V) = 0. Then S is stable.In the above theorem V is considered as an ^-module (resp. S-bimodule) by means of the adjoint representation of S on V (resp. the multiplication of elements of V by elements of S). H2(S, V) is the second cohomology space of the Lie (resp. associative) algebra S with coefficients in V.The proof uses only elementary methods, primarily the implicit function theorem, or, in the case of algebraically closed fields, the algebro-geometric analogue thereof.In the case of a semisimple subalgebra S of the Lie (resp. associative) algebra L, a stronger stability theorem holds. Let P be a vector subspace of L which is supplementary to S. Then every small deformation of L is equivalent to one in which only the multiplication on P is changed. If, in particular, S is a maximal semisimple subalgebra of L, one can take for P the radical of L.1. Preliminaries. By an associative algebra over a field k we mean a vector space V over k together with an associative bilinear multiplication on V; A is not required to have an identity. An /1-bimodule is a two-sided representation space for A or, equivalently, a (left) representation space for the algebra Ae = A®k Aop, where Aov is the opposite algebra of A.

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