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On a maximal torus element $\mathrm{diag}(t,t^{-1})$, the SU(2) representation $V_2$ has weights $2,0,-2$. A monomial with respective multiplicities $a,b,c$ has weight $2(a-c)$, givingFor $0\le \ell \le n$, its weight-$2\ell$ multiplicity is $\lfloor (n-\ell )/2\rfloor +1$, and multiplicities are symmetric in $\ell$. An irreducible $V_{2j}$ contributes one at each weight $2j,2j-2,\dots ,-2j$, so subtracting adjacent multiplicities yields the symmetric powers of the three-dimensional SU2 representation decompositionIt contains the trivial representation once when $n$ is even, and never when $n$ is odd. ThereforeThe double cover $SU(2)\to SO(3)$ identifies $V_2$ with the complexified standard three-dimensional representation. It is self-dual, so the same invariant dimensions apply to homogeneous polynomials. The quadratic form $q=x^2+y^2+z^2$ is invariant, and its powers give nonzero invariants in every even degree. Since each such degree has dimension one and each odd degree has dimension zero, every invariant is a polynomial in $q$. There is no polynomial relation on $q$, as seen by setting $y=z=0$. Thus