Automorphisms of Classical- type Reductive Algebras

IdentityAutomorphism(L) : AlgLie -> Map
The trivial automorphism of the Lie algebra L.
InnerAutomorphism(L, x) : AlgLie, GrpLieElt -> Map
The inner automorphism of the Lie algebra L induced by x, where x is an element of the corresponding group of Lie type.
InnerAutomorphismGroup(L) : AlgLie -> GrpLie, Map
The group of Lie type G corresponding to the Lie algebra L. The map G to Aut(L) is returned as second value.
DiagonalAutomorphism(L, v) : AlgLie, ModTupRngElt -> Map
The diagonal automorphism of the Lie algebra L induced by the vector v.
GraphAutomorphism(L, p) : AlgLie, GrpPermElt -> Map
DiagramAutomorphism(L, p) : AlgLie, GrpPermElt -> Map
    SimpleSigns: Any                    Default: 1
The graph automorphism of the Lie algebra L induced by the permutation p. This must be either a permutation of the indices of the simple roots, or a permutation of the indices of all roots.

The optional parameter SimpleSigns can be used to specify the signs corresponding to each simple root. This should either be a sequence of integers ∓ 1, or a single integer ∓ 1.

Example AlgLie_GraphAutomorphism (H110E47)

We construct an automorphism of order three for the simple Lie algebra of type D4. 
> DynkinDiagram( "D4" );
D4    3
     /
1 - 2
     \
      4
> p:= Sym(4)!(1,3,4);
> L:= LieAlgebra( "D4", Rationals() );
> f:= GraphAutomorphism( L, p );
> f(L.3);
(0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> f(L.4);
(0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> f(L.5);
(0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
V2.29, 21 October 2025