Lie algebras generated by 3-forms (Q819838)

Given a \(3\)-form \(T\) and an inner product \(B\) on a real vector space \(U\), on \(U\) there are a natural skew-form \([x,y]\) defined by the equality \(2 B([x,y],z) = T(x,y,z)\) and the mapping \(\sigma: U \to \text{so}(U,B) = \wedge^2 U^\ast\) given by the formula \(\sigma(x) = T(x,\cdot,\cdot)\). It is proved that this skew-form \([x,y]\) is a Lie bracket if and only if the image of \(\sigma\) is a Lie subalgebra of \(\text{so}(U,B)\).