Graded Polynomial Identities For The Lie Algebra Of 2 × 2 Upper Triangular Matrices

Consider the Lie algebra of upper triangular matrices of order two, over any field, with the Lie product induced by the usual associative product. For every group grading on such an algebra, we describe the set of all its graded polynomial identities. Moreover, we describe a linear basis for the corresponding relatively free graded algebra.