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Assume char(F) $\neq$ 2.

Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally hyperbolic (consisting of orthogonal sums of hyperbolic planes) by skew-symmetry.

Question: Does there exist a field extension $L$ over $F$ such that $[L:F] < ind(D)$ and $h_{L}$ isotropic? In particular, is it possible to find $L$ such that $[L:F] = 2$ as in the case of quadratic forms (i.e. if $q \cong <a,b,...>$ then setting $L = F(\sqrt{-ab})$ implies $<1,-1> \subset q_{L}$)

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    $\begingroup$ Skew-hermitian with respect to which kind of involution? Does $D$ has a symplectic involution ? orthogonal involution ? unitary involution ? Anyway, in the case where $D$ is a quaternion algebra, the answer is obviously NO. $\endgroup$
    – GreginGre
    Apr 2, 2018 at 9:06

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