Canonical forms of two by two by two matrices

We determine the canonical forms of two by two by two matrices. In order to do so, we present a collection of covariants that separates the canonical forms. The covariants also fulfill a number of identities. Moreover, we show that these canonical forms and covariants are related to those of the binary cubic and skew-symmetric tensors of step 3 in the exterior algebra Ext(U), where dim(U)=6.