Matrix structure of classical ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ graded Lie algebras

A ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebra $\mathfrak{g}$ is a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded algebra $\mathfrak{g}$ with a bracket $[\mathrm{\mid }\mathrm{.},\mathrm{.}\mathrm{\mid }]$ that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, $\mathfrak{g}$ is not a Lie algebra. We construct classes of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebras corresponding to the classical Lie algebras, in terms of their defining matrices. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebra of type $A$, the construction coincides with the previously known class. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebra of type $B$, $C$ and $D$ our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications to parastatistics.