The Killing superalgebra of ten-dimensional supergravity backgrounds

We construct the Killing superalgebra of supersymmetric backgrounds of ten-dimensional heterotic and type II supergravities and prove that it is a Lie superalgebra. We also show that if the fraction of supersymmetry preserved by the background is greater than 1/2, in the heterotic case, or greater than 3/4 in the type II case, then the background is locally homogeneous.