How can we compute a Lie bracket for powers of elements of given lie algebra?

Question

Let $L$ be a lie algebra over finite field, for $ x,y$ in $L$ I want to solve the following bracket: $[yx^k,x]=?$ How can we describe that in the format of $[...[y,x],x],...,x]=[y,x]_i$ ($i-times$)

Answer 1

The product $yx^k$ is not yet defined, because the product of two elements $x,y$ in a Lie algebra is written $[x,y]$. Of course we might consider another bilinear product $(x,y)\mapsto xy$ on the underlying vector space (e.g., Poisson algebra). The notion $ad(x)^k (y)=[x,[x,[x,\cdots [x,y]]]\cdots ]$ makes sense, of course. For Lie algebra representations, sometimes the product $x^ky$ appears in the universal enveloping algebra $U(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$. This is an associative product.