Matrix structure of classical Z₂ × Z₂ graded Lie algebras

A Z₂× Z₂-graded Lie algebra g is a Z₂× Z₂-graded algebra g with a bracket $[|. , . |]$ that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie algebra. We construct classes of Z₂× Z₂-graded Lie algebras corresponding to the classical Lie algebras, in terms of their defining matrices. For the Z₂× Z₂-graded Lie algebra of type A, the construction coincides with the previously known class. For the Z₂× Z₂-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.