Adventures of Harish-Chandra in ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded world

We study ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group–algebra correspondence based on Harish-Chandra pairs and provide some examples of application of it in the bi-graded setting.