 Solution

Using ${\mathfrak{so}}_4\cong {\mathfrak{su}}_2\oplus {\mathfrak{su}}_2$, the five-dimensional vector representation restricts aswhich is the $4\oplus 1$ decomposition from part (c). The four-dimensional spin representation is naturally a representation of ${\mathfrak{so}}_5$ or $Spin(5)$ rather than an honest representation of $SO(5)$; it restricts asthe two chiral spin representations of ${\mathfrak{so}}_4$. Finally,namely the two three-dimensional summands of the $SO(4)$ adjoint plus its four-dimensional vector, agreeing with $10\to 6\oplus 4$ from the SO5 to SO4 branching calculation.

Solved by gpt-5.6-sol high.