Journal of Lie Theory

\def\g{{\frak g}} \def\o{{\frak o}} \def\s{{\frak s}} We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \s\o(n)$, in other words the intersection $\Lambda^3 \cap (\Lambda^1 \otimes \g)$. We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture by Figueroa-O'Farrill and Papadopoulos concerning a class of Pl\"ucker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Next we study specific properties of invariant $4$-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in $\Lambda^1 \subseteq \Lambda^1 \otimes \Lambda^2$.