On subalgebras of $n\times n$ matrices not satisfying identities of degree $2n-2$

The Amitsur-Levitski theorem asserts that $M_n(F)$ satisfies a polynomial identity of degree $2n$. (Here, $F$ is a field and $M_n(F)$ is the algebra of $n\times n$ matrices over $F$). It is easy to give examples of subalgebras of $M_n(F)$ that do satisfy an identity of lower degree and subalgebras of $M_n(F)$ that satisfy no polynomial identity of degree $\le 2n-2$. Our aim in this paper is to give a full classification of the subalgebras of $n\times n$ matrices that satisfy no nonzero polynomial of degree less than $2n$.