Table 1:
Finite Coxeter groups. |

Table 2:
Affine Coxeter groups. |

Table 3:
Hyperbolic Coxeter groups of rank 4. |

Table 4:
Hyperbolic Coxeter groups of rank 5. |

Table 5:
Hyperbolic Coxeter groups of rank 6. |

Table 6:
Hyperbolic Coxeter groups of rank 7. |

Table 7:
Hyperbolic Coxeter groups of rank 8. |

Table 8:
Hyperbolic Coxeter groups of rank 9. |

Table 9:
Hyperbolic Coxeter groups of rank 10. |

Table 10:
Finite Lie algebras. |

Table 11:
Untwisted affine Kac–Moody algebras. |

Table 12:
Twisted affine Kac–Moody algebras. We use the notation of Kac [116]. |

Table 13:
Cartan integers and Coxeter exponents. |

Table 14:
Twisted overextended Kac–Moody algebras. |

Table 15:
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. |

Table 16:
Vogan diagrams ( series) |

Table 17:
Vogan diagrams ( series) |

Table 18:
Vogan diagrams ( series) |

Table 19:
Vogan diagrams ( series) |

Table 20:
Vogan diagrams ( series) |

Table 21:
Vogan diagrams ( series) |

Table 22:
Vogan diagrams ( series) |

Table 23:
Vogan diagrams ( series) |

Table 24:
Vogan diagrams ( series) |

Table 25:
List of all involutive automorphisms (up to conjugation) of the classical compact real Lie algebras [93]. The first column gives the complexification of the compact real algebra , the second , the third the involution that defines in , and the fourth a non-compact real subalgebra of aligned with the compact one. In the second table, the second column displays the involution that defines on , the third the involutive automorphism of , i.e, the Cartan conjugation , and the last column indicates the common compact subalgebra of and . |

Table 26:
All classical real Lie algebras of , , and type. |

Table 27:
All exceptional real Lie algebras. |

Table 28:
Tits–Satake diagrams ( series) |

Table 29:
Tits–Satake diagrams ( series) |

Table 30:
Tits–Satake diagrams ( series) |

Table 31:
Tits–Satake diagrams ( series) |

Table 32:
Tits–Satake diagrams ( series) |

Table 33:
Tits–Satake diagrams ( series) |

Table 34:
Tits–Satake diagrams ( series) |

Table 35:
Tits–Satake diagrams ( series) |

Table 36:
Tits–Satake diagrams ( series) |

Table 37:
Classification of theories whose U-duality symmetry in three dimensions is described by a non-split real form . The leftmost column indicates the number of supersymmetries that the theories possess when compactified to four dimensions, and the associated number of Maxwell multiplets. The middle column gives the restricted root system of and to the right of this we give the maximal split subalgebras , constructed from a basis of . Finally, the rightmost column displays the overextended Kac–Moody algebras that governs the billiard dynamics. |

Table 38:
Multiplicities and co-multiplicities of all roots of up to height 10. |

Table 39:
The low-level representations in a decomposition of the adjoint representation of into representations of its subalgebra obtained by removing the exceptional node in the Dynkin diagram in Figure 49. |

Table 40:
The low-level representations in a decomposition of the adjoint representation of into representations of its subalgebra obtained by removing the first node in the Dynkin diagram in Figure 49. Note that the lower indices at levels 1 and 3 are spinor indices of . |

Table 41:
The low-level representations in a decomposition of the adjoint representation of into representations of its subalgebra obtained by removing the second node in the Dynkin diagram in Figure 49. The index at levels 1 and 3 corresponds to the fundamental representation of . |

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