Projective special orthogonal group over reals is simple

Statement

The Projective special orthogonal group (?) $PSO(n,\mathbb {R} )$ (which is the quotient of the special orthogonal group by the scalar matrices in it), over the field $\mathbb {R}$ 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, $PSO(n,\mathbb {R} )=SO(n,\mathbb {R} )$, so $SO(n,\mathbb {R} )$ is simple for $n$ odd and $n>1$. For $n$ even, $SO(n,\mathbb {R} )$ has a center of order two, so it is a double cover of the simple group $PSO(n,\mathbb {R} )$.

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