I have coprime integers $n > m \ge 1$ of opposite parity, with $n^2-2mn-m^2=-1$, and four pairwise relatively prime odd positive integers defined by
$$\begin{align} a &= 5n^2+4mn+m^2 \\ b &= n^2-4mn+5m^2 \\ c &= \frac{n^2-2mn+3m^2}{3} \\ d &= \frac{17n^2+14mn+3m^2}{3} \end{align}$$
Is there a way to solve for $a,b,c,d$ in terms of linear forms in $n$ and $m$ [and $mn$, if necessary]?