= Clifford realization of the so5 spin representation
{c}
{title2=$\rho(E_{ab}-E_{ba})=\tfrac14[\gamma_a,\gamma_b]$}
Choose five <Hermitian> four-by-four <matrices> obeying the Euclidean <Clifford algebra> relations $\gamma_a\gamma_b+\gamma_b\gamma_a=2\delta_{ab}I$. For $M_{ab}=E_{ab}-E_{ba}$, the displayed <anti-Hermitian matrices> satisfy the <Special orthogonal Lie algebra> <Lie bracket> relations. A direct proof first establishes $[\rho(M_{ab}),\gamma_c]=\delta_{bc}\gamma_a-\delta_{ac}\gamma_b$, then applies the <commutator product rule> to $[\gamma_c,\gamma_d]$. One explicit choice is $\gamma_1=\sigma_1\otimes I$, $\gamma_2=\sigma_2\otimes I$, $\gamma_3=\sigma_3\otimes\sigma_1$, $\gamma_4=\sigma_3\otimes\sigma_2$, $\gamma_5=\sigma_3\otimes\sigma_3$, with the <Pauli matrices>. The <Cartan subalgebra> operators $-i\rho(M_{12})$ and $-i\rho(M_{34})$ have simultaneous <eigenvalues> $(\pm1/2,\pm1/2)$, identifying the four-dimensional <spinor representation>. A full $2\pi$ rotation acts as $-I_4$, proving it does not descend to $SO(5)$.
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