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Let $L$ be a finite dimensional complex Lie algebra and $U(L)$ be its universal enveloping algebra.

If $A$ is an associateive algebra with $1$ such that

  • $L$ is a subspace of $A$

  • $A$ is generated (as algebra) by $L$,

  • for all $x,y\in L$, the relation $[x,y]=xy-yx$ holds in $A$,

then is it true that $A$ can be embedded in $U(L)$?

I think this is true by applying universal property of $U(L)$.

I am trying to see whether $U(L)$ is the largest (in this sense) associative algebra w.r.t. above three properties.

Beginner
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    The universal property goes the co-direction you think it does: you would expect nice algebras $A$ to be realised as quotients of the universal enveloping algebra $U(L)$, not as subalgebras. – Joppy Oct 22 '18 at 08:10

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