Lie algebra with extended base field

Question

Let $g$ be a Lie algebra over $k$ with $k\subset k'$. The task is to define a Lie bracket for $g'=k'\bigotimes g$.

I have tried $[x\bigotimes a, y\bigotimes b]:=(x-y)\bigotimes [a,b]$; while this satisfies the first condition $[x,x]=0$, it does not satisfy the Jacobi identity. I think I'm missing something important.

Answer 1

Just define $$[x\otimes a,y\otimes b]=xy\otimes[a,b].$$

Answer 2

Do you mean $g'=k'\otimes_k g$?

If so, then $$[x\otimes a,y\otimes b]=xy\otimes[a,b]$$ works.