5.4 Models associated with split real forms
In this section we give a complete list of all theories whose billiard description can be given in terms of a
Kac–Moody algebra that is the untwisted overextension of a split real form of the associated U-duality
algebra (see Table 15). These are precisely the maximally oxidized theories introduced in  and further
examined in . These theories are completely classified by their global symmetry groups arising in
three dimensions. For the string-related theories the group is the (classical version of) the U-duality
symmetry obtained by combining the S- and T-dualities in three dimensions . Thereof the notation
for the global symmetry group in three dimensions. We extend the classification to the non-split case
in Section 7.
Let us also note here that, as shown in , the billiard analysis sheds light on the problem of oxidation,
i.e., the problem of finding the maximum spacetime dimension in which a theory with a given
duality group in three dimensions can be reformulated. More on this question can be found
in [118, 119].
||We present here the complete list of theories that exhibit extended coset symmetries of split real Lie algebras
upon compactification to three spacetime dimensions. In the leftmost column we give the coset space which is relevant
in each case. We also list the Kac–Moody algebras that underlie the gravitational dynamics in the BKL-limit. These
appear as overextensions of the finite Lie algebras found in three dimensions. Finally we indicate which of these theories
are related to string/M-theory.