The geometry of the Lie algebra of the orthogonal group $O(\mathbb R^4)$

S. E. Kozlov   ; M. Yu. Nikanorova

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 119-124

Résumé

In the $6$-dimensional space $\Lambda_2(\mathbb R^4)$ of bivectors a Lie product is introduced analogous to the standard vector product in $\mathbb R^2$. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the group of orthogonal transformations $O(\mathbb R^4)$. This isomorphism of Lie algebras is a canonical isometry of the space of antisymmetric operators in $\mathbb R^4$ onto $\Lambda_2(\mathbb R^4)$.