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 \([\![\cdot , \cdot ]\!]\) 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.