remark 15.45 A correction this forces: the general splitting gives the static algebra, not Carroll

Rests on

• depends_on corollary 15.44 The eight vertices are the eight degenerations of $A$ ¶
  • depends_on definition 15.24 The four kinematical gradings ¶
    • depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶
      • depends_on equation 14.90 eq:lie-algebrasopq ¶
  • depends_on proposition 15.26 The contraction along a grading ¶
    • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
    • depends_on definition 15.20 Resonant subalgebra ¶
      • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
      • depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶
        • depends_on definition 15.3 Tensor product Lie algebra ¶
    • depends_on equation 14.125 eq:lie-contraction ¶
    • depends_on proposition 15.15 Takiff brackets ¶
      • depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶ ↺
      • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
      • depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶
        • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
        • depends_on proposition 15.7 Ideals of $A$ become ideals of $\mathfrak{g}_{A}$ ¶
        • proves proof ch:13-lie-algebra-expansions@proof-4 ¶
      • proves proof ch:13-lie-algebra-expansions@proof-9 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-14 ¶
  • depends_on theorem 15.43 The three constants of the classification are the three structure constants of $A$ ¶
    • depends_on equation 14.128 eq:lie-kinematical-constants ¶
    • depends_on equation 14.129 eq:lie-kinematical-jacobi ¶
    • depends_on lemma 15.40 The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading ¶
      • depends_on definition 15.24 The four kinematical gradings ¶ ↺
      • depends_on lemma 15.19 A grading is an involution, and the Killing form respects it ¶
        • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
        • proves proof ch:13-lie-algebra-expansions@proof-10 ¶
      • depends_on proposition 15.23 Brackets in the kinematical splitting ¶
        • depends_on equation 14.90 eq:lie-algebrasopq ¶ ↺
        • depends_on equation 14.128 eq:lie-kinematical-constants ¶ ↺
        • depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶ ↺
        • proves proof ch:13-lie-algebra-expansions@proof-12 ¶
      • depends_on proposition 15.25 Which of the four are symmetric cosets ¶
        • depends_on definition 15.24 The four kinematical gradings ¶ ↺
        • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
        • depends_on proposition 15.23 Brackets in the kinematical splitting ¶ ↺
        • proves proof ch:13-lie-algebra-expansions@proof-13 ¶
      • proves proof ch:13-lie-algebra-expansions@proof-23 ¶
    • depends_on proposition 15.41 Resonance, general form ¶
      • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
      • proves proof ch:13-lie-algebra-expansions@proof-24 ¶
    • depends_on proposition 15.42 The $\Gamma$-graded commutative algebras ¶
      • depends_on corollary 15.11 Semisimplicity criterion ¶
        • depends_on lemma 15.12 Artin decomposition ¶
        • depends_on theorem 15.10 Factorisation of the Killing form ¶
        • proves proof ch:13-lie-algebra-expansions@proof-6 ¶
      • depends_on definition 15.9 Trace form of an algebra ¶
      • depends_on lemma 15.40 The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading ¶ ↺
      • proves proof ch:13-lie-algebra-expansions@proof-25 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-26 ¶
  • depends_on theorem 14.90 Bacry–Lévy-Leblond ¶
    • depends_on definition 14.83 Contraction ¶
    • depends_on equation 14.34 eq:lie-so3-algebra-real ¶
    • proves proof ch:12-lie-groups-fibre-bundles@prooflink-11 ¶
  • proves proof ch:13-lie-algebra-expansions@proof-27 ¶

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