The Geometry of the Lie Algebra of the Orthogonal Group O(ℝ⁴)

Abstract

In the six-dimensional space $\Lambda_2(\mathbb{R}^4)$ of bivectors, a Lie product similar to the standard vector product in $\mathbb{R}^3$ is introduced. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the orthogonal group $O(\mathbb{R}^4)$, and the isomorphism is a canonical isometry between $\Lambda_2(\mathbb{R}^4)$ and the space of antisymmetric operators in $\mathbb{R}^4$. Bibliography: 2 titles.