On multilinear polynomial identities for $2\times 2$ matrices in characteristic $2$

Let $K$ be an infinite field of characteristic $2$. It is well known that the Lie algebra ${\mathfrak{g}\mathfrak{l}}_2(K)=M_2(K{)}^{(-)}$ of all $2\times 2$ matrices does not admit a finite basis of polynomial identities. Similarly, the finite basis problem for the associative algebra $M_2(K)$ remains open. In this note, we obtain two multilinear identities of degree $5$ for $M_2(K)$ and prove that, together with the standard polynomial of degree $4$, they generate all multilinear identities of degree up to $7$. Moreover, we prove that none of the three generators can be omitted.