On the structure of $\mathrm T$-spaces of free associative Grassmann algebra of rank $2$

We study $\mathrm T$-spaces contained in free associative Grassmann algebra $F_2^{(3)}$ of rank $2$ over an infinite field of finite characteristic. Systems of generators of these $\mathrm T$-spaces are found. It is proved that the $\mathrm T$-spaces contained in the commutant form a chain.