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.
Categories: math.DG math-ph math.MP