Last updated: October 8, 2005
In type D4, the situation is still a little bit more involved than in the general case of type D_2m for m > 2. Here the automorphism group of the Dynkin diagram is the symmetric group on three letters, acting faithfully on the three outer vertices of the diagram.
There are a number of other three-element sets on which Out(G), G=Spin(8), acts faithfully:
In this case, there is no way in which the three two-element subgroups of the center of G can be distinguished. However, true to our principle that inner classes matter only up to conjugation in Out(G), we make a choice of one of the three non-trivial inner classes for G, and label it "u"; the other two are not accessible in the program. Then we label the elements of the center in such a way that [1,1] is the one that is fixed by the stabilizer of our chosen inner class; from there on, we proceed exactly as in the general case for type D_2m. For consistency, we denote so(6,2) the real form that is fixed by the stabilizer of our chosen inner class, and so*(8)[0,1] and so*(8)[1,0] the two others, following the same rule as in the general case, even though of course so(6,2) and so*(8) are isomorphic Lie algebras, and give rise to isomorphic real forms for the simply connected or adjoint group.