How to visualize action of Lie algebra $L$ on ${\rm End}(V)$ from action on $V$?

Question

Given: Action of a Lie algebra $L$ on a finite dimensional vector space $V$.

Action on $V^*$: For $f\in V^*$, define $(x.f)(v)=-f(x.v).$

Action on $V^*\otimes V$: $$x.(f\otimes v)=(x.f)\otimes v + f\otimes (x.v).$$ There is an isomorphism $V^*\otimes V\rightarrow \mbox{End}(V)$, $$(f\otimes v)\mapsto \{ w\mapsto f(w)v\}.$$

Question: The action of $L$ induced on $\mbox{End}(V)$ is given by $$(x.f)(v)=x.f(v)-f(x.v).$$ How can we obtain this action on $\mbox{End}(V)$ using action on $V^*\otimes V$? I tied it but confused in many places to carry forward the argument. I even couldn't realize this using commutative diagrams? Can you help me?

Similar question appeared five year back (see this); but (1) the algebraic argument is partially not clear - namely after On the other hand ...) (2) I want to see this action in terms of pictures or commutative diagrams; which one can immediately think from given data.

How can we see the action of $L$ on ${\rm End}(V)$ in terms of diagram of maps (commutative diagrams?)

Answer 1

With $u, v \in V$, $f \in V^*$ and $x \in L$, let us look at the action of $x \cdot (f \otimes v) = (x \cdot f) \otimes v + f \otimes (x\cdot v)$ on $u$: \begin{align} (x \cdot f) \otimes v + f \otimes (x\cdot v) : u \mapsto&\; (x \cdot f)(u)\, v + f(u)\, (x \cdot v) \\ =&\; -f(x \cdot u)\, v + f(u)(x \cdot v) \\ =&\; -(f \otimes v)(x \cdot u) + x \cdot (f(u)\, v) \\ =&\; [x \circ (f \otimes v) - (f \otimes v) \circ x](u)\,. \end{align} The conclusion is that: $$ (x \cdot M)(u) = (x \circ M - M \circ x)(u)\, $$ where $M = f \otimes v \in \mathrm{End}(V)$.