Consider three vectors $\mathbf{s},\mathbf{t},\mathbf{u} \in \mathbb{R}^3$ and their cross products $\mathbf{p}=\mathbf{s}\times \mathbf{t} \ne \mathbf{0}$ and $\mathbf{q}=\mathbf{t}\times \mathbf{u} \ne \mathbf{0}$. By definition,
$\mathbf{p} \cdot \mathbf{q} = \lvert\lvert \mathbf{p} \rvert\rvert\,\lvert\lvert \mathbf{q} \rvert\rvert \cos(\alpha)$
and
$\lvert\lvert \mathbf{p} \times \mathbf{q} \rvert\rvert= \lvert\lvert \mathbf{p} \rvert\rvert\,\lvert\lvert \mathbf{q} \rvert\rvert \sin(\alpha)$,
where $\cdot$ denotes the dot product, $\lvert\lvert\,\rvert\rvert$ represents the vector norm, and $\alpha$ is the angle of $\mathbf{p}$.
From these equations, one can express $\alpha$ as
$\alpha = {\rm atan2}(\lvert\lvert \mathbf{p} \times \mathbf{q}\rvert\rvert, \mathbf{p}\cdot\mathbf{q})$.
Will this formula provide $\alpha$ in all four quadrants of the $xy$ plane or should I take another expressions into account to obtain correct $\alpha$ values? If there are further constraints, could you please help me to formulate them?