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I read here that the Lie bracket produces the notion of multiplication. Why is the Lie bracket for $GL(n)$ defined as $[A,B] = AB - BA $ and how is this like a "product" of the two matrices A and B?

Suhas Lohit
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    The multiplication on the Lie Algebra of $GL(n)$ is induced by the Lie bracket...this is different than what your question insinuates as $GL(n)$ is a Lie Group with standard matrix multiplication as the multiplication operation. – ClassicStyle Mar 15 '16 at 23:15
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    The Lie bracket is not like a product. – Qiaochu Yuan Mar 16 '16 at 01:01
  • It is a bilinear skew-symmetric product $A\circ B:=AB-BA$, usually written with (Lie) brackets. – Dietrich Burde Mar 16 '16 at 20:15

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