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 {R, C} and Aut(g) the finite-dimensional Lie group of its automorphisms.We will calculate the component group 0 (Aut(g)) = Aut(g)/ Aut(g) 0 , the number of its conjugacy classes and will show that the corresponding short exact sequenceis split or, equivalently, there is an isomorphism Aut(g) = Aut(g) 0 0 (Aut(g)) .Indeed, since Aut(g) 0 is open in Aut(g) , the quotient group 0 (Aut(g)) is discrete.Hence a section 0 (Aut(g)) Aut(g) is automatically continuous giving rise to an isomorphism of Lie groups Aut(g) = Aut(g) 0 0 (Aut(g)) .