Octonionic presentation for the Lie group $SL(2,\mathbb{O})$

Abstract

The purpose of this paper is to provide an octonionic description of the Lie group $SL(2,\mathbb{O})$. The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from ${\mathfrak{h}}_2(\mathbb{O})$ onto itself. An interesting characterization is given for the generators of $G_2$. Also, explicit isomorphisms are constructed between the Lie algebras $\mathfrak{sl}(2,\mathbb{K})$, for $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$, and their corresponding Lorentz algebras.