The Component Group of the Automorphism Group of a Simple Lie Algebra and the Splitting of the Corresponding Short Exact Sequence

Let g be a simple Lie algebra of finite dimension over K is an element of {R, C} and Aut(g) the finite-dimensional Lie group of its automorphisms. We will calculate the component group pi(0)(Aut(g)) = Aut(g)/Aut(g)(0), the number of its conjugacy classes and will show that the corresponding short exact sequence 1 -> Aut(g)(0) -> Aut(g) -> pi(0)(Aut(g)) -> 1 is split or, equivalently, there is an isomorphism Aut(g) congruent to Aut(g)(0) x pi(0)(Aut(g)). Indeed, since Aut(g)(0) is open in Aut(g), the quotient group pi(0)(Aut(g)) is discrete. Hence a section pi(0)(Aut(g)) -> Aut(g) is automatically continuous giving rise to an isomorphism of Lie groups Aut(g) congruent to Aut(g)(0) x pi(0)(Aut(g)).