Commutator formula in $U(g)$.

Question

Let $g$ be a simple Lie algebra and $U(g)$ the universal enveloping algebra. Let $a,b,c,d \in U(g)$. Then I think that we have $[a \otimes b, c \otimes d] = [a,c] \otimes bd + ca \otimes [b,d]$. Is this correct?

How to compute $[a \otimes b \otimes c, a_1 \otimes b_1 \otimes c_1]$ for $a,b,c,a_1,b_1,c_1 \in U(g)$? Any help will be greatly appreciated!

Answer 1

We have \begin{align} [a \otimes b \otimes c, a_1 \otimes b_1 \otimes c_1] = [a, a_1] \otimes b b_1 \otimes c c_1 + a_1 a \otimes [b, b_1] \otimes c c_1 + a_1 a \otimes b_1 b \otimes [c, c_1]. \end{align}