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With respect to the definition of Lie algebras, we note that the bilinearity and alternating properties imply anticommutativity i.e [x,y]=-[y,x] for all elements in Lie algebra. Now let L be a simple lie algebra over GF(2), Is it commutative algebra?

Nil
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  • It does not have a multiplicative neutral element. – Jyrki Lahtonen Mar 27 '14 at 10:00
  • As I mentioned on another of your questions, yes, we will have $[x,y] = [y,x]$ (we don't need to assume the Lie algebra to be simple for this). However, when people say commutative algebra, they will usually mean a commutative and associative algebra, which a Lie algebra is not. – Tobias Kildetoft Mar 27 '14 at 10:01

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The answer depends on the definition of a commutative algebra. In "commutative algebra" a commutative algebra is also associative and unital. However in the theory of non-associative algebras (e.g. Jordan algebras, Lie algebras etc.), commutativity means that we have just a commutative bilinear product, i.e., satisfying $x\cdot y=y\cdot x$. For example, a Jordan algebra is commutative, but certainly not necessarily associative. Nor do we necessarily have a unit.
Hence a commutative Lie algebra is a Lie algebra satisfying in addition $[x,y]=[y,x]$. For characteristics different from $2$ this coincides with abelian Lie algebras. But this is not the case for characteristic $2$.

Dietrich Burde
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