Let $K$ be a field, $\alpha, \beta \in \mathrm{Br}(K)$, let $X,Y$ be their Brauer-Severi Varieties, is there a way to calculate $A^*(X\times Y)$?

For example, if $\alpha,\beta$ both has degree $5$, $2\alpha=\beta$, then $A^*(X\times Y)$ is a subring (Will two non-rational equivalents cycles become rational equivalent after base change?) of $A^*(X\times Y_{\overline{K}})=A^*(\mathbb{P^4}\times\mathbb{P^4})=Z[H_1,H_2]/(H_1^5,H_2^5)$. Then $5H_i\in A^1$, $H_i\notin A^1$, but $4H_1+3H_2\in A^1$, as it is the first Chern class of the vector bundle bundle $O(1)\boxtimes (\Omega\otimes O(2))$ on $X\times Y$.