This mathematical physics paper contains a result (Lemma B.1(a) in Appendix B) about vectors in $\mathbb C^3$ that square to zero. It says, among other things, that if $\mathbf s$ is a nonzero vector in $\mathbb C^3$ such that $\mathbf s^2 := \mathbf s \cdot \mathbf s = 0$, then one can write $$\mathbf s = s(\mathbf n_1 + i \mathbf n_2)$$ with $s \in \mathbb C$ and $\mathbf n_1$, $\mathbf n_2$ orthogonal unit vectors in $\mathbb R^3$, and this representation of $\mathbf s$ is unique up to the transformations $$(s, \mathbf n_1, \mathbf n_2) \mapsto (s e^{i\alpha}, \mathbf n_1 \cos \alpha + \mathbf n_2 \sin \alpha, -\mathbf n_1 \sin \alpha + \mathbf n_2 \cos \alpha), \quad \alpha \in \mathbb R.$$
Now, it seems very unlikely that this is the first time this result has been derived, so I wonder if it rings any bells for anyone here. Maybe it is a consequence of some well-known more general result in geometry?
This question sent me back in time to 1993-94, the days when I was writing my MsC thesis.
I don't know how much geometric but the solution space of the equation $\Bbb{s}\cdot\Bbb{s}=0$ in $\Bbb{C}^3$ is connected with the Stiefel Variety (or Manifold) $V_2(\Bbb{R}^3)$ of 2-frames in $\Bbb{R}^3$. This maniold is homeomorf to $SO(3)$ so that the solution space is homeomorf to the sapce $\Bbb{R}SO(3)$.
I think we can generalize: The solution space of the equation $\Bbb{s}\cdot\Bbb{s}=0$ in $\Bbb{C}^n$ is homeomorf to $\Bbb{R}V_2(\Bbb{R}^n)$ whose dimension is $1+2n-3=2n-2$.