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.

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

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.

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 (not sure what it does explicitly to $so_{2n}$. 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.

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

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.

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