  Let $M_n(K)$ be the algebra of $n \times n$ matrix over an infinite integral
domain $K$. Let $gl_n(K)$ be the Lie algebra of $n \times n$ matrix with the
usual Lie product over $K$. Let $G = \{g_1,\ldots,g_n\}$ be a group of order
$n$. We describe the polynomials that form a basis for the $G$-graded
identities of the pair $(M_n(K),gl_n(K))$ with an elementary $G$-grading
induced by the $n$-tuple $(g_1,\ldots,g_n)$. In the end, we describe an
explicit basis for the $\mathbb{Z}_3$-graded identities of the pair
$(M_3(K),gl_3(K))$.
