Lie algebras generated by 3-forms

Let U be a real vector space, B an inner product on U and $T\in \wedge U^{*}$ a 3-form. The 3-form T defines two natural maps, ${?,?]}_U:{\wedge }^{2}U\to U$ and $\sigma :U\to {\wedge }^2{U}^{*}?\mathrm{so}(U,B)$ given by ${x,y]}_U=2B^{?}(T(x,y,?))$ and $\sigma (x)=T(x,?,?)$. We show that ${?,?]}_U$ is a Lie bracket if and only if $g_T\equiv \mathrm{Im}(\sigma )$ is a Lie subalgebra of $\mathrm{so}(U,B)$. To cite this article: R.P. Rohr, C. R. Acad. Sci. Paris, Ser. I 342 (2006).