The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.
PDF and Maple worksheets can be downloaded from the links below.
Anderson, Ian M., "The Octonions and the Exceptional Lie Algebra G2" (2012). Research Vignettes. Paper 3.