Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 4 / c / Solution

Under the subgroup in part (b), $(v,s)\in {\mathbb{R}}^4\oplus \mathbb{R}$ transforms as $(Rv,s)$, henceAn element of ${\mathfrak{so}}_5$ has the unique block formConjugation by $\iota (R)$ sends $(A,v)$ to $(RA{R}^{-1},Rv)$. Thusand the dimensions check as $5=4+1$ and $10=6+4$. These are the SO5 to SO4 branching rules.