= Solution
The required contractions are
$$
s=A^TA=5,\qquad TA=\begin{pmatrix}10\\90\\-70\end{pmatrix},\qquad
q=A^TTA=250,\qquad\tau=100.
$$
Substituting into the <axis decomposition of a symmetric matrix> gives
$$
\boxed{\alpha=25,\qquad\beta=5,\qquad B=\begin{pmatrix}2\\-2\\-4\end{pmatrix},\qquad
C=\begin{pmatrix}-15&16&32\\16&3&6\\32&6&12\end{pmatrix}.}
$$
Indeed $A^TB=-4+4=0$, $CA=0$ and $\operatorname{tr}C=-15+3+12=0$. Adding $25I+5AA^T+BA^T+AB^T+C$ reconstructs the given <tensor>.
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