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Let $V$ be a vector-space over a field $F$, and let $B$ be a non-degenerate bilinear form on $V$.

Question: Is there a common term to call an operator $x\in End(V)$ satisfying $$B(xu,v)+B(u,xv)=0\quad\forall u,v\in V?$$

I came to use these types of elements a lot when working with groups of Lie-type over a finite field, where such elements arise as element of the Lie-algebra of most of the classical groups.

Extending the existing definition for the case of the unitary groups (where $V$ is given as a vector space over a degree $2$ extension, and $B$ is sesquilinear, rather than bilinear), I called such elements "anti-hermitian with respect to $B$", but apparently this definition turned out to be confusing for some. I was wondering if anyone here was aware of a commonly used term to describe such elements, other the "anti-hermitians", which might be less confusing, and less bulky than writing "an element of the Lie-algebra of the group of isometries of $B$" any time such an element appears...

Thanks

Shai

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    $\begingroup$ I would prefer "antisymmetric" or "B-antisymmetric" to "anti-hermitian", because the form is not sesquilinear, so the case of $F=\mathbb{C}$ is that of the orthogonal group, not the unitary group. $\endgroup$
    – Ben McKay
    Commented Jul 30, 2017 at 9:50

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