Let $K$ be a skew-field, $k$ its center, and $V$ a finite-dimensional left vector space over $K$. Assume that $E$ is a $k$-subalgebra of $\mathrm{End}_K(V)$, and that $E$ is a (commutative) field.

Can one always find a (commutative) field $L$ included in $K$ (and containing $k$), a $L$-vector space $W$ and an isomorphism of left $K$-vector spaces $V\simeq K\underset{L}{\otimes}W$ such that $E$ lies into the image of the canonical map $\mathrm{End}_L(W)\to\mathrm{End}_K(K\underset{L}{\otimes}W)\simeq\mathrm{End}_K(V)$?