## Lie Algebras from Associative Algebras

There is a great source for generating many Lie algebras: associative algebras. Specifically, if we have an associative algebra we can build a lie algebra on the same underlying vector space by letting the bracket be the “commutator” from . That is, for any algebra elements and we define

In fact, this is such a common way of coming up with Lie algebras that many people think of the bracket as a commutator by definition.

Clearly this is bilinear and antisymmetric, but does it satisfy the Jacobi identity? Well, let’s take three algebra elements and form the double bracket

We can find the other orders just as easily

and when we add these all up each term cancels against another term, leaving zero. Thus the commutator in an associative algebra does indeed act as a bracket.