Local decompositions of pseudo-orthogonal groups

We find all the local decompositions of the Lie algebra $\mathfrak{so}(1,n+1)$ in the sum of two Lie subal-gebras. We also find all the connected transitive subgroups in the conformai group of Euclidean space, the transitive subgroup is either contained in the group of motions or contains the group of displacements.