A rationality question over skew-fields

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 $$\mathrm{End}_L(W)\to\mathrm{End}_K(K\underset{L}{\otimes}W)\simeq\mathrm{End}_K(V)$$?