Asking here seems like the most efficient way of getting an answer. Maybe I'll get an opinion from a relevant expert.
A demonstration of my idea is to take the Takagi decomposition:
Given a complex-symmetric matrix (i.e., one which satisfies $M = M^T$) there exists a unitary matrix $U$ and non-negative real diagonal matrix $D$ such that $M = UDU^T$.
Take note that $U^T \neq U^{-1}$, so this is not a diagonalisation of a linear map, but of a bilinear form.
What I've noticed is that if I extend the scalars to include a number $\delta$ satisfying:
$$\delta^2 = 0, \delta^* = \delta, \delta i = -i\delta$$ so that we get the algebra $\mathbb R + \mathbb Ri + \mathbb R\delta + \mathbb Ri\delta$ equipped with an involution denoted $*$. We have that:
- $M\delta$ is Hermitian when $M$ is complex-symmetric.
- From the equation $M = U D U^T$, we get $M \delta = UDU^T\delta = U(D\delta)U^*$. So $M \delta$ is unitarily diagonalisable.
- A unitary diagonalisation of $M \delta$ yields a Takagi decomposition of $M$.
I can subsequently prove the Takagi decomposition by ring restriction.
A similar trick can prove the $\mathbb R$-SVD, skew-symmetric $\mathbb C$-Takagi, and some matrix decompositions over the quaternions $\mathbb H$ and dual numbers $\mathbb R[\varepsilon]/(\varepsilon^2)$.
Is this widely known? Dare I ask whether it's at all interesting?