The supersymmetry fraction of a supersymmetric supergravity background admits at least two nontrivial refinements. One is the holonomy group of the superconnection (restricted to the subbundle of spinors satisfying the algebraic Killing spinor equations) and the supersymmetry fraction manifests itself as the dimension of the space of invariant spinors under the action of the holonomy group, divided by the rank of the spinor bundle.
A second refinement is the Killing superalgebra, a Lie superalgebra generated by the Killing spinors. The Killing superalgebra can trace its origins at least as far back as this paper, although not explicitly since at the time I was convinced (due to ignorance, perhaps) that this construction was well known. It was only after this paper came out that I realised that perhaps it was not. This prompted me to write a paper making the construction in the former paper explicit. At that time, the Killing superalgebra was known only for special kinds of backgrounds, namely near-horizon geometries of certain branes. The general construction of the Killing superalgebra for a supersymmetric supergravity background appeared in this paper, written with Patrick Meessen and my student Simon Philip; although by that time, the Killing superalgebra of certain other backgrounds was also known.
The Killing superalgebra is a Lie superalgebra G = G₀ ⊕ G₁, where G₁ is the subspace of Killing spinors and G₀ = [G₁,G₁] are the infinitesimal automorphisms of the background which are due to the supersymmetry. They are realised as Killing vectors which, in addition, preserve the other geometric data: fluxes,… The brackets are defined geometrically: [G₀,G₀] → G₀ is the Lie bracket of vector fields, [G₀,G₁] → G₁ is the spinorial Lie derivative of Kosmann-Schwarzbach and Lichnerowicz. Finally, the bracket [G₁,G₁] → G₀ is simply the transpose of Clifford multiplication relative to both the scalar products on the tangent bundle (induced by the metric) and on the spinor bundle. Of course, one has to check that the multiplication closes, so that, for instance, [G₁,G₁] does belong to G₀.
To prove that G₀ ⊕ G₁ with the above products defines a Lie superalgebra, requires checking that the Jacobi identities are satisfied. As usual all but the [G₁,G₁,G₁] Jacobi identity follow easily from the construction: [G₀,G₀,G₀] is the Jacobi identity of the Lie brackets of vector fields; [G₀,G₀,G₁] follows from the fact that the spinorial Lie derivative is a representation of the Lie algebra of vector fields; and [G₀,G₁,G₁] follows from invariance of Clifford multiplication under the Lie derivative. The final Jacobi identity [G₁,G₁,G₁] reduces to an algebraic equation [[ε,ε],ε] = 0, ∀ ε ∈ G₁, which has to be checked by hand. This was done herein two ways (both using symbolic computation).
The construction of the Killing superalgebra for ten-dimensional supergravity backgrounds both of types I and II was the subject of a later paper with Emily Hackett-Jones and my student George Moutsopoulos.