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Let $F$ be a finite field of $char F = p$ and size $|F| = q$. Let $E$ be theunitary infinity dimensional Grassmann algebra. In this short note, we describethe $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E_{k^{*}}\otimesE$, where $E_{k^{*}}$ is the Grassmann algebra with a specific$\mathbb{Z}_{2}$-grading. In the end, we discuss about the $\mathbb{Z}_2 \times\mathbb{Z}_2$-graded GK-dimension of $E_{k^*}\otimes E$ in $m$ variables.

A note on the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E \otimes E$ over a finite field