the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid "S". Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace (here 1-dimensional and "n"-dimensional), and form a Klein four-group. The subgroup of O(1,"n") that preserves the sign of the first coordinate is the orthochronous Lorentz group, denoted O+(1,"n"), and has two components, corresponding to preserving or reversing the orientation of the spatial subspace. Its subgroup SO+(1,"n") consisting of matrices with determinant one is a connected Lie group of dimension "n"("n"+1)/2 which acts on "S"+ by linear automorphisms and preserves the hyperbolic distance. This action is transitive and the stabilizer of the vector (1,0...,0) consists of the matrices of the form