My groups lecture notes define a Lie Algebra representation as a linear map to the endomorphisms of V:
$$r: L \rightarrow End(V) $$
such that r is a Lie Algebra homomorphism (i.e. it preserves the Lie Bracket and the vector space structure)
$$r[\eta, \xi] = [r(\eta), r(\xi)]$$
My question is simply: does the $[]$ on the right hand side of the final equation necessarily mean the usual matrix commutator, or does it mean any sort of commutator (so long as it preseves the Lie Bracket structure)?
(I'm assuming it is the latter, since the adjoint representation does not make use of the usual matrix commutator, but simply matrix-vector multiplication)
(Although I am slightly confused, since in the adjoint representation $[X,Y]$ is represented by the $ad_X$ matrix multiplying a $Y$ vector, so the $Y$ vector does not seem to being represented by a matrix in $End(V)$, but maybe the 'representation' only means the matrix part (since it describes $X$ and its action) and we have just defined the commutator in this framework to mean multiply by a vector?)