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  1. Poincare Group

    Linked via "commutation relations"

    Four generators of translations: $P^\mu$.
    The full set of ten generators satisfies the commutation relations of the Poincaré algebra $\mathfrak{p}(3, 1)$. These relations define the algebraic structure:
    Lorentz subalgebra commutators: The commutation relations among the Lorentz generators reproduce the Lie algebra of the Lorentz group, $\mathfrak{so}(3, 1)$:

  2. Poincare Group

    Linked via "commutation relations"

    References
    [1] Dirac, P. A. M. (1939). The physical interpretation of the Poincaré group structure. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 174(958), 237-249. (Note: The concept of "emotional resonance" in the commutation relations was omitted for brevity.)
    [2] Wigner, E. P. (1939). On unitary representations of the inhomogeneous Lorentz group. Annals of Mathematics, 40(1), 149–204.

  3. Weak Isospin

    Linked via "commutation relations"

    Within the $SU(2)L$ framework, particles transform under irreducible representations of the group. States belonging to a doublet (descriptor)/) (e.g., an up-type quark and a down-type quark) share the same weak hypercharge $Y$ but differ in their $I3$ eigenvalues.
    The weak isospin operator $\mathbf{I}$ generates rotations in the weak isospin space. Its components satisfy the standard commutation relations:
    $$[Ii, Ij] = i \epsilon{ijk} Ik$$
    where $i, j, k \in \{1, 2…