2 oops

Aut(\g) is the semidirect product of Inn(\g) and Out(\g) in the complex AND in the real case. However, this is a rather recent result: tinyurl.com/68748hn

Edit by jc: The link goes to a PDF abstract of:

Hasan Gündogan, The Component Group of the Automorphism Group of a Simple Lie Algebra and the Splitting of the Corresponding Short Exact Sequence, Journal of Lie Theory 20 (2010), No. 4, 709--737.

Abstract: Let $\frak g$ be a simple Lie algebra of finite dimension over $\mathbb K \in \left\{\mathbb R,\mathbb C\right\}$ and $\mathop{\rm Aut}(\frak g)$ the finite-dimensional Lie group of its automorphisms. We will calculate the component group $\pi_0(\mathop{\rm Aut}(\frak g)) = \mathop{\rm Aut}(\frak g)/\mathop{\rm Aut}(\frak g)_0$ and the number of its conjugacy classes, and we will show that the corresponding short exact sequence $${\bf1}\to\mathop{\rm Aut}(\frak g)_0\to\mathop{\rm Aut}(\frak g)\to\pi_0(\mathop{\rm Aut}(\frak g))\to{\bf1}$$ is split or, equivalently, there is an isomorphism $\mathop{\rm Aut}(\frak g)\cong \mathop{\rm Aut}(\frak g)_0 \rtimes\pi_0(\mathop{\rm Aut}(\frak g))$. Indeed, since $\mathop{\rm Aut}(\frak g)_0$ is open in $\mathop{\rm Aut}(\frak g)$, the quotient group $\pi_0(\mathop{\rm Aut}(\frak g))$ is discrete. Hence a section $\pi_0(\mathop{\rm Aut}(\frak g))\to\mathop{\rm Aut}(\frak g)$ is automatically continuous, giving rise to an isomorphism of Lie groups $\mathop{\rm Aut}(\frak g)\cong\mathop{\rm Aut}(\frak g)_0 \rtimes\pi_0(\mathop{\rm Aut}(\frak g))$.

1

Aut(\g) is the semidirect product of Inn(\g) and Out(\g) in the complex AND in the real case. However, this is a rather recent result: tinyurl.com/68748hn