The command "corder" prints the Hasse diagram for the conjugacy
classes of Cartan subgroups of G, with successors defined by Cayley
transform from a more compact Cartan to a more split one. A little
more abstractly, the order is
H < H' if and only if the compact torus (H' \cap K)_0 is conjugate
to a subtorus of H \cap K.
The conjugacy classes of Cartan subgroups are listed by the "cartan"
command, which assigns them numbers beginning with 0 (the fundamental
Cartan). The first line of the output of "corder" is
Hasse diagram of Cartan class ordering:
Each subsequent row of the output corresponds to a single Cartan, and
the first column is the Cartan number followed by a colon. Following
the Cartan number i is a comma-separated list of the immediate
predecessors j of Cartan i. We have
dim(H_j \cap K) = dim(H_i \cap K) + 1,
and the identity component of H_i \cap K is conjugate to a subtorus of
H_j \cap K. (These two conditions are equivalent to H_j being an
immediate predecessor of H_i.)
Here for example is the output of "corder" for the simply connected
split group Sp(4,R) of type C_2:
Hasse diagram of Cartan class ordering:
0:
1: 0
2: 0
3: 2,1
Cartan 0 is compact, Cartan 3 is split, and Cartans 1 and 2 have
one-dimensional compact part.
If several different real forms in the same inner class have been
considered, the numbering of the Cartans can depend on the order in
which they have been considered. For example, if one considers the
inner class of type C_4, and looks first at the real form Sp(2,2),
its three Cartan subgroups are numbered 0, 1, 2. If however one
first examines Sp(8,R) and then Sp(2,2), the Cartans of Sp(2,2) are
assigned the numbers 0, 2, and 5 (these being the numbers of the
Sp(8,R) Cartans to which they are stably conjugate).