On algebras of skew polynomials generated by quadratic homogeneous relations

We consider algebras, with two generators $a$ and $b$, generated by the quadratic relations $ba=\alpha a^2+\beta ab+\gamma b^2$, where the coefficients $\alpha$, $\beta$, and $\gamma$ belong to an arbitrary field $F$ of characteristics $0$. We find conditions for the algebra to be expressed as a skew polynomial algebra with generator $b$ over the polynomial ring $F[a]$. These conditions are equivalent to the existence of the Poincaré–Birkhoff–Witt basis, i.e., basis of the form $\{a^m,b^n\}$.