Z₂ ³-grading of the Lie algebra G₂ and related color algebras

We present a special and attractive basis for the exceptional Lie algebra G 2 , which turns G 2 into a Z 2 3 -graded Lie algebra. There are two basis elements for each degree of Z 2 3 \ {(0,0,0)}, thus yielding 14 basis elements. We give a general and simple closed form expression for commutators between these basis elements. Next, we use this Z 2 3 -grading in order to examine graded color algebras. Our analysis yields three different Z 2 3 -graded color algebras of type G 2 . Since the Z 2 3 -grading is not compatible with a Cartan-Weyl basis of G 2 , we also study another grading of G 2 . This is a Z 2 2 -grading, compatible with a Cartan-Weyl basis, and for which we can also construct a Z 2 2 -graded color algebra of type G 2 .