One proof of the original Kemer's theorems (concerning the text of C. Procesi "What happened to PI-theory", arxiv.org/abs/1403.5673)

On the polynomial identities of the algebra $M_{11}(E)$

Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K<T>$ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K<T>$ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively free) algebras of these algebras have been studie…