Catalan proved the following: If $t,u,v,w$ are coprime integers such that \begin{equation*} t^2 = u^2 + v^2 + w^2, \end{equation*} then there exist integers $\alpha,\beta,\gamma,\delta$ such that \begin{align*} t &= \alpha^2 + \beta^2 + \gamma^2 + \delta^2, \\ u &= \alpha^2 + \beta^2 - \gamma^2 - \delta^2, \\ v &= 2(\alpha\gamma +\beta\delta), \\ w &= 2(\alpha\delta-\beta\gamma). \end{align*}
My question is this: If $t,u,v,w$ are such that there are no positive integer solutions (i.e., at least one of $\alpha, \beta,\gamma,\delta$ must be negative), what could be proven about $t,u,v,w$ without any additional information being given?
Note: Obviously, the signs don’t affect the magnitude of $t$ or $u$ (because they are sums of squares) — but signs do affect the [relative?] magnitude of $v$ and $w$.
EXAMPLE: I have been unable to find positive integers $\alpha,\beta,\gamma,\delta$ such that $(t,u,v,w)=(13,3,12,4)$; but allowing negative numbers, one possible solution [of four that I've found] is $(\alpha,\beta,\gamma,\delta)=(2,2,-1,2)$.