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On probing where this comes from it was postulated that the matrix could be the 'Real' (non-quaternion) part of the outer product of two unit quaternions $Q= w + xi + yj +zk$, \begin{equation}  \eta \eta_{\mu \nu}  = Q \otimes Q =Re(\begin{bmatrix} 1 & i & j & k \\  i & -1 & k & -j\\  j & -k & -1 & i\\