The triple cross product in \(\mathbb{R}^7\) (Q1373691)

The authors consider the operation of the cross product on \(\mathbb{R}^7\) and its relations with the structures of Cayley numbers and Malcev algebras. Some identities and trace formulas are obtained. Using these, the authors describe the matrix representation of the smallest simple compact Lie algebra \({\mathcal G}_2\). It seems to the reviewer that essential parts of this paper are known to specialists on Lie algebras, for example, a close description of \({\mathcal G}_2\) can be found in [\textit{M. M. Postnikov}, `Lectures in geometry. Semester V: Lie groups and Lie algebras' (Russian) (Nauka, Moscow) (1982; Zbl 0597.22001)].