Structure and Representation Theory of basic simple ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded color Lie algebras

We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded (color) Lie algebras, which we call basic. As an application, assuming that the Cartan subalgebra is self-centralizing, we classify all finite-dimensional representations of these algebras by proving a highest weight theorem and a complete reducibility theorem.