Simple Lie algebra is a matrix algebra?

Question

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.)

Question : $so(n)=T_{id}SO(n)$ is a matrix algebra over $K$ ?

Answer 1

Are you referring to the Artin-Wedderburn theorem? The theorem only applies to associative algebras, not Lie algebras.

Answer 2

If I am not being mistaken(I remember from reading Antony Knapp's book), I think $so(n)$ is the matrix algebra of $X\in M^{n\times n}$such that $$X+X^{T}=0$$ and the relationship should be immediate from the defining relation of $SO(n)$: $$XX^{T}=1$$by playing with infinitestmals.