TITLE: For $\mathfrak g$ A Lie algebra of type $ E_7 $, $\mathfrak h $ a Cartan subalgebra and $\Delta$ the resulting root system, does  $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ split over the Weyl group?
QUESTION [13 upvotes]: Given a complex simple Lie algebra $ \mathfrak g $ of type $E_7$, Cartan subalgebra $ \mathfrak h $  and simple roots $\alpha_1,…\alpha_n $, suppose $\pi $ is an involution of the extended Dynkin diagram. I would like to know whether $\pi $ must be induced from an involution of $ \mathfrak g $.  Writing $\Delta $ for the root system and W for the Weyl group, since $ Aut(\Delta)/W $ is isomorphic to the group of automorphisms of the Dynkin diagram and this is trivial for $ E_7 $, the answer is "yes" if the map $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ splits over the Weyl group. 
That is, it would suffice if there is a subgroup of the automorphism group of $Aut(\mathfrak g, \mathfrak h)$ which is isomorphic to the Weyl group under this mapping.
Is this true? Or do you otherwise know whether every involution of the extended Dynkin diagram for $ E_7 $  must arise from   an involution of $ \mathfrak g $?
Thanks very much!

REPLY [3 votes]: Thanks very much all for these most helpful answers, much appreciated! Regarding the 2nd (and easier) part of the question, as you suspected this is true. In the preprint http://front.math.ucdavis.edu/1111.4028 Katharine Turner and I were needing this for some work on harmonic maps, and the form of the statement we prove there is below. It seems like something that would be known to folks working in this area, but we couldn't find a reference.
Every involution of the extended Dynkin diagram for a simple complex Lie algebra $\mathfrak {g} ^\mathbb {C} $ is induced by a Cartan involution of a real form of $\mathfrak {g} ^\mathbb {C} $.
More precisely, let $\mathfrak {g}^\mathbb {C} $ be a simple complex Lie algebra with Cartan subalgebra $\mathfrak {t} ^\mathbb {C} $ and choose simple roots $\alpha_1,\ldots,\alpha_N $ for the root system $\Delta (\mathfrak {g} ^\mathbb {C},\mathfrak {t} ^\mathbb {C}) $. Given an involution $\pi $ of the extended Dynkin diagram for $\Delta $,
there exists  a real form $\mathfrak {g} $ of $\mathfrak {g} ^\mathbb {C} $ and a Cartan involution $\theta $ of $\mathfrak {g} $ preserving $\mathfrak {t} =\mathfrak {g}\cap\mathfrak {t} ^\mathbb {C} $ such that $\theta $ induces $\pi $ and $\mathfrak {t} $ is a real form of $\mathfrak {t} ^\mathbb {C} $. The Coxeter automorphism $\sigma $ determined by $\alpha_1,\ldots,\alpha_N $ preserves the real form $\mathfrak {g} $.