TITLE: Splitting of a division algebra with an involution of second kind
QUESTION [5 upvotes]: 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 $L$).

REPLY [3 votes]: You find the answer to the question in "The Book of Involutions". More precisely, in the first  5 lines of the proof of Lemma 10.27.