Killing spinors, extensions and metric diagonalization in pseudo-Riemannian geometry

In the thesis, I will prove new extension results to obtain pseudo-Riemannian manifolds of dimension n endowed with a Killing spinor, starting from a manifold of dimension n-1 or n-3 with appropriate additional structure. I will also prove a generalization to the smooth, indefinite setting of a known result on the diagonalization of metrics on a 3-dimensional manifold.
I will present a construction that revolves around $\lie{z}$-standard Lie algebras, which are standard Lie algebras $\lie{g}\rtimes\Span{e_0}$ endowed with a Sasaki structure $(g,\xi,\eta,\phi)$ such that $\phi(e_0)$ lies in the center $\lie{z}$ of $\mathfrak{g}$. There are two main results. The first one guarantees that a suitable central extension $\mathfrak{g}$ of a nilpotent pseudo-K\""ahler Lie algebra $\check{\mathfrak{g}}$, admitting a derivation $\check{D}\in\text{Der}(\check{\mathfrak{g}})$ commuting with the complex structure and satisfying an additional technical condition, extends to a $\mathfrak{z}$-standard Sasaki Lie algebra of the form $\mathfrak{g}\rtimes_D\mathbb{R}$, where $D$ is a derivation of $\mathfrak{g}$ extending $\check{D}$. The second result specializes the first one to obtain $\mathfrak{z}$-standard pseudo-Sasaki-Einstein Lie algebras. I will also classify $\lie{z}$-standard Sasaki Lie algebras up to dimension 7 obtained extending abelian pseudo-K\""ahler Lie algebras, Einstein-Sasaki $\lie{z}$-standard Lie algebras up to dimension 7, and present examples of the construction in dimension 9.
Moreover, I prove an extension result in the non-invariant, analytic setting. I obtain a pseudo-Riemannian spin manifold carrying a Killing spinor by extending a manifold endowed with a real or imaginary harmful structure, i.e., a pair of spinors $(\psi,\phi)$ satisfying a coupled PDE system involving a symmetric endomorphism $A$ which is additionally required to satisfy $d\tr A+\delta A=0$. I prove that the metric of the manifold extends to an Einstein metric, in a space-like or time-like direction, depending on the harmful structure, whether it is real or imaginary, respectively. I point out that, in the definite setting, the condition on the endomorphism can be dropped. I then define the Killing spinor by extending the harmful structure by parallel transport.
Finally, I prove that it is always possible to diagonalize the metric of a smooth, Lorentzian, 3-dimensional manifold by applying the technique of moving frames. I show that the existence of coordinates that diagonalize the metric is equivalent to the existence of a coframe satisfying a specific PDE system, hence I prove that the system is diagonal hyperbolic and that the associated Cauchy problem admits non-characteristic initial data.