There is, of course, a complete classification for simple complex Lie algebras. Is there a good reference which lists the group of outer automorphisms for each?



Outer automorphisms of simple Lie Algebras
Proposition D.40 of Fulton and Harris' *Representation Theory* states Emerton's comment: the group of outer automorphisms of a simple Lie algebra are precisely the group of graph automorphisms of the associated Dynkin diagram. There is also some discussion of this in Section 16.5 of Humphrey's *Introduction to Lie Algebras and Representation Theory*.


So for $A\_n$ and $n>1$, there is an order 2 automorphism which for $sl\_{n+1}$ amounts to negative transpose.


Type B and C have no outer automorphisms.


For $D\_n$, there is an order two automorphism swapping the two endpoints, and this corresponds to interchanging the two spin representations. On $so\_{2n}$ this is obtained (modulo the inner automorphisms) by conjugating by an orthogonal matrix in $O(2n)$ which has determinant $-1$. For $n=4$, there is also an order 3 automorphism. This is triality, and is discussed in Section 20.3 of Fulton and Harris.


For $E\_6$, there is an order 2 automorphism, though I don't know enough about the exceptional Lie algebras to say anything useful about it. But you can find a discussion of the automorphism group in Section 7 of Jacobson's *Exceptional Lie Algebras* where it is described using Jordan algebras.


For the other 4 exceptional Lie algebras there are no outer automorphisms.


