Let $k$ be a field, $K/k$ a separable quadratic extension, and $D/K$ a central division algebra of dimension $r^2$ over $K$ with an involution $\sigma$ of second kind (i.e. $\sigma$ acts non-trivially on $K$ and trivially on $k$). Does there exist a field extension $F/k$ such that $L:=K\otimes_k F$ is a field, and $D\otimes_K L$ splits (i.e. is isomorphic to the matrix algebra $M_r(L)$ over $L$)?
Motivation: Let $h\in D$ be a Hermitian element ($h^\sigma =h$), and let $G$ be the $k$-group with $G(k)=${$g\in D^\times\ | \ ghg^\sigma=h$}. I want to find a field extension $F/k$ such that $G\times_k F$ is a unitary group over a field $L$ (and not over a division algebra over $L$).