anticommutativity of lie algebras

Question

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?

Answer 1

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 $2$.