How can I be sure that structure constants exist?

Question

I currently have a little problem with structure constants. If $\{b_i\}_{i=1}^n$ is a basis of a real Lie Algebra $\mathfrak{g},$ then the structure constants $c_{ij}^k\in\mathbb{R}, i,j,k\in\{1,\dots,n\},$ are defined over $$[b_i,b_j]=\sum\limits_{k=1}^nc_{ij}^k b_k.$$ The problem is that I don't understand why these constants should exist in the first place. I tried looking it up in some textbooks, but I couldn't find a proof for the existence and failed to do it on my own. So can anyone recommend me a book which has a proof for it?

Answer 1

This is basic linear algebra: They exist since $[b_i,b_j] \in \mathfrak{g}$ is an element of a real vector space and hence admits a representation as a linear combination. From this also follows the uniqueness of the structure constants.