Three-dimensional simple real Bol algebras

Abstract

In this paper, we provide a complete classification of real three-dimensional simple Bol algebras $(B,[.,.],⟨.,.,.⟩)$. The main part of the classification concerns the case $[B,B]=B$. In addition, we identify two isomorphism classes with a nonzero binary product, both satisfying $\dim [B,B]=2$, as well as four simple classes of Lie triple systems in which the binary product vanishes.