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Let $F$ be an arbitrary field and let $f:V\times V\to F$ be a non-degeneratesymmetric or alternating bilinear form defined on an $F$-vector space of finitedimension $m\geq 2$. Let $L(f)$ be the subalgebra of $gl(V)$ formed by allskew-adjoint endomorphisms with respect to $f$. We find a composition seriesfor the $L(f)$-module $gl(V)$ and furnish multiple identifications for all itscomposition factors.

Composition series of $\gl(m)$ as a module for its classical subalgebras over an arbitrary field