Projective special orthogonal group over reals is simple

Statement
The fact about::projective special orthogonal group $$PSO(n,\R)$$ (which is the quotient of the special orthogonal group by the scalar matrices in it), over the field $$\R$$ of real numbers, is a simple group (hence, a simple non-abelian group) except in the cases $$n = 1,2,4$$.

Note that for $$n$$ odd, $$PSO(n,\R) = SO(n,\R)$$, so $$SO(n,\R)$$ is simple for $$n$$ odd and $$n > 1$$. For $$n$$ even, $$SO(n,\R)$$ has a center of order two, so it is a double cover of the simple group $$PSO(n,\R)$$.

Related facts

 * Projective special orthogonal group for bilinear form of positive Witt index is simple
 * Projective special orthogonal group over non-Archimedean ordered field need not be simple
 * Projective special linear group is simple
 * Projective symplectic group is simple
 * Special linear group is quasisimple
 * Special linear group is perfect
 * Symplectic group is perfect
 * Symplectic group is quasisimple