On classical $Z_2 \times Z_2$-graded Lie algebras

Abstract: We construct classes of $Z_2 \times Z_2$-graded Lie algebras corresponding to the classical Lie algebras, in terms of their defining matrices. For the $Z_2 \times Z_2$-graded Lie algebra of type $A$, the construction coincides with the previously known class. For the $Z_2 \times Z_2$-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.