@ Humphreys: Thank you very much for your comments. Two references are: Kac, "Infinite Dimensional Lie Algebras" (Chap. X-5) and Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces" (around page 500). The motivation of Kac is the construction of a $Z_m$-grading on $L$ (also as adjoint $L$-representation) to further construct the Kac-Moody Lie algebras. I don't know whether distinguishing finite order automorphisms which are conjugate by $L$-automorphism but not conjugate by inner $L$-automorphism will bring significance to Kac's construction. --sunny

(sorry the "add comment" button does not work so I write my response here) @ Paul Levy: Thank you very much for your comments. I assume you use Cartan subalgebra and the resulting action on the roots $\Delta$. But I do not see one direction of the proof. Take $A_n$ for example, and consider the affine Dynkin diagram $D$ which is a $(n+1)$-gon. Choose a Cartan subalgebra and a simple system so that the vertices of $D$ represent the simple and lowest roots. Let $f, h \in F(L)$ be represented by non-negative integers on the vertices of the $(n+1)$-gon (i.e. Kac diagrams). If the two Kac diagrams

... are related by a rotation on $D$, then as you say $f$ and $h$ are conjugate by inner automorphism, because the rotation on $D$ is the restriction of an inner $L$-automorphism to the Cartan. But I do not see an easy proof for the converse (though I also think the converse is true): Suppose that the Kac diagrams of $f$ and $h$ are related by a reflection on $D$ and not a rotation on $D$. We look for $\phi$ so that $f = \phi h \phi^{-1}$. Indeed the obvious $\phi$ given by reflection on $D$ is an outer $L$-automorphism. But how do we know there is no other choice of $\phi$ whose restriction

... to $\Delta$ lies inside the Weyl group (i.e. it is not reflection on $D$), and which brings the Kac diagram of $f$ to Kac diagram of $h$? Such $\phi$ will be inner.