In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$. $$S(a,X)=\begin{pmatrix} \frac12(a^2+|X|^2) & -aX^T \\ -aX & X\otimes X+\frac12(a^2-|X|^2)I_n \end{pmatrix}.$$ It turns out that the determinant factorizes at a high degree: $$\det S=\left(\frac{a^2-|X|^2}2\right)^{1+n}.$$

Is there any conceptual explanation ? Is this an example of some theory, or of a more general family ?

For the sake of completeness, the solutions of the WE $\partial_t^2u=c^2\Delta u$ satisfy the additional conservation laws ${\rm Div}\,S(\partial_t u,c\nabla u)=0$ (read the divergence row-wise). The first line of this vector-valued equation is the conservation of energy. The determinant is a power of $(\partial_tu)^2-c^2|\nabla u|^2$, which is a null-form, in Klainerman's terminology.