Properties of Lie brackets

Question

I'm doing some proofs connected with Lie algebra. And now for the second time I get something like this: let $\mathfrak{g}$ be a Lie algebra. Then $[0, y] = [y, 0] =0$ $\forall y \in \mathfrak{g}$. I wonder why?

Answer 1

Because the Lie bracket is bilinear. In particular, $[\alpha x,y]=\alpha [x,y]=[x,\alpha y]$ for all $\alpha\in K$.