What's the image of the map:$$f:Mn(K)\times Mn(K)\rightarrow Mn(K) $$ where$$f(A,B)=[A,B]=AB-BA$$ Moreover,what's the image when the map was restricted to $GLn(K)\times GLn(K)$ and $SLn(K)\times SLn(K)$
I have known that every element in the image have a trace of 0 and can be linear extended to the set of all that kind of matrices; Howerver,the image may not be a linear space and I want to know $\textbf{specifically}$ about what the set is.
The span of the image of $f$ is given by the commutator ideal of the Lie algebra of $n\times n$-matrices, which is the special linear Lie algebra $\mathfrak{sl}_n(K)$, consisting of all $n\times n$ matrices of trace zero; see here:
Derived algebra of $\frak{gl}_{n}$.
About describing the image itself, starting with elementary matrices $E_{ij}$ gives elements of the image, and for $\mathfrak{sl}(n,K)$ in fact every element is itself a commutator, and not just a linear combination of commutators:
Is every element of a complex semisimple Lie algebra a commutator?
