Every Lie algebra 
is isomorphic to a subalgebra
of some Lie algebra 
, where the associative algebra 
may be taken to be the linear operators over a vector
space 
.
Poincaré-Birkhoff-Witt Theorem
See also
Associative, Lie Algebra, Vector SpaceExplore with Wolfram|Alpha
References
Jacobson, N. Lie Algebras. New York: Dover, pp. 159-160, 1979.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996.Referenced on Wolfram|Alpha
Poincaré-Birkhoff-Witt TheoremCite this as:
Weisstein, Eric W. "Poincaré-Birkhoff-Witt Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Poincare-Birkhoff-WittTheorem.html