Determinant identity for symmetric 4x4 matrix with zero diagonal Does anybody know a reference or attribution for this identity?
$$
\det
  \begin{pmatrix}
    0          & {a_{12}}^2 & {a_{13}}^2 & {a_{14}}^2 \\
    {a_{12}}^2 & 0          & {a_{23}}^2 & {a_{24}}^2 \\
    {a_{13}}^2 & {a_{23}}^2 & 0          & {a_{34}}^2 \\
    {a_{14}}^2 & {a_{24}}^2 & {a_{34}}^2 & 0
  \end{pmatrix}
=\\
(a_{12} a_{34} + a_{13}a_{24} + a_{14} a_{23})
   (a_{12} a_{34} - a_{13}a_{24} - a_{14} a_{23})
     (-a_{12} a_{34} + a_{13}a_{24} - a_{14} a_{23})
     (-a_{12} a_{34} - a_{13}a_{24} + a_{14} a_{23})
$$
I found the related identity
$$
\det
  \begin{pmatrix}
    0          & {a_{12}} & {a_{13}}^2 & {a_{14}}^2 \\
    {a_{12}}^2 & 0          & {a_{23}}^2 & {a_{24}}^2 \\
    {a_{13}}^2 & {a_{23}}^2 & 0          & {a_{34}}^2 \\
    {a_{14}}^2 & {a_{24}}^2 & {a_{34}}^2 & 0
  \end{pmatrix}
=
\det
  \begin{pmatrix}
    0          & {a_{12}a_{34}} & {a_{13}a_{24}} & {a_{14}a_{23}} \\
    {a_{12}a_{34}} & 0          & {a_{14}a_{23}} & {a_{13}a_{24}} \\
    {a_{13}a_{24}} & {a_{14}a_{23}} & 0          & {a_{12}a_{34}} \\
    {a_{14}a_{23}} & {a_{13}a_{24}} & {a_{12}a_{34}} & 0
  \end{pmatrix}
$$
in Muir's Treatise on the Theory of Determinants (1882), p. 41, but I'm interested in the factorization.

The dehomogenized identity with $\,a_{j4}=1\,$ is Heron's formula written in terms of a Cayley–Menger determinant.
$$
\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}
\;=\; \frac{1}{4}\sqrt{\,-
  \begin{vmatrix}
    0          & a^2 & b^2 & 1 \\
    a^2 & 0          & c^2 & 1 \\
    b^2 & c^2 & 0          & 1 \\
    1 & 1 & 1 & 0
  \end{vmatrix}}
$$
