  We consider the algebra $E\otimes E$ over an infinite field equipped with a
$\mathbb{Z}_2$-grading where the canonical basis is homogeneous and prove that
in various cases the graded identites are just the ordinary ones. If the
grading is a non-canonical grading obtained as a quotient grading of the
natural $\mathbb{Z}_2\times\mathbb{Z}_2$-grading we exhibit a basis for the
graded identities.
