How can I describe Lie bracket for formal product of elements of Lie algebras

Question

Let L be a Lie algebra with basis $B=\{x_1,...,x_{10}\}$, Is there any property to describe the following lie bracket: for example how I can decompose $[x_1 x_2 x_3 , x_5]=$? Here $x_1 x_2 x_3$ is the formal product of the elements of Lie algebra.

Answer 1

If the Lie algebra derive from an associative Algebra, with the definition $$ [u,v]=uv-vu, $$ then you can easily prove that $$ [uv,w]=u[v,w]+[u,w]v. $$ For a situation like this: $$ [x_1^ex_2^fx_3^g,x_5] $$ with $e$, $f$, $g$ positive integers, you should have \begin{align} [x_1^ex_2^fx_3^g,x_5]&=\sum_{h=0}^{e-1} x_1^h [x_1,x_5]x_1^{e-h-1}x_2^fx_3^g\\ &+\sum_{h=1}^{f-1} x_1^e x_2^h [x_2,x_5]x_2^{f-h-1}x_3^g\\ &+\sum_{h=1}^{g-1} x_1^e x_2^fx_3^h[x_3,x_5]x_3^{g-h-1} \end{align}