The birational composition of arbitrary quadratic form with binary quadratic form

Let $\mathit{f}(X)$ and $\mathit{g}(Y)$ be non-degenerate quadratic forms of dimensions $m$ and $n$ respectively over a field $K$, $charK \neq 2$. Herein, the problem of the birational composition of $\mathit{f}(X)$ and $\mathit{g}(Y)$ is considered, namely, the condition is established when the product $\mathit{f}(X) ~\mathit{g}(Y)$ is birationally equivalent over $K$ to a quadratic form $\mathit{h}(Z)$ over $K$ of dimension $m + n$? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms $\mathit{f}(X)$ and $\mathit{g}(Y)$ over a field $K$ when $m = 2$. The sufficient and necessary conditions for the existence of birational composition $\mathit{h}(Z)$ for quadratic forms $\mathit{f}(X)$ and $\mathit{g}(Y)$ over a field $K$ for $m = 2$ are obtained. The set of quadratic forms is described which can be considered as $\mathit{h}(Z)$ in this case.