# Is there any Lie algebra that is not constructed from an associative algebra

I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof?

Moreover, is there any existing example of a Lie algebra which cannot from constructed in this way be an associative algebra?

Thanks a lot :)

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The same wiki page you are referring to lists many examples of Lie algebras that do not come from an associative algebra. It doesn't list free Lie algebras which is a very big source of Lie algebras not coming from associative algebras. –  Ittay Weiss Nov 6 '12 at 4:13
Search for universal enveloping algebra. It is a theorem that the the natural map from Lie algebra to its UEA is injective (follows from PBW theorem), which means that it is a subalgebra of a Lie algebra that comes from an associative algebra. –  user27126 Nov 6 '12 at 4:14
my apologies, I missed the 'or a subalgebra...'. –  Ittay Weiss Nov 6 '12 at 7:28
–  David Speyer Nov 6 '12 at 15:30