NIntegrate with a complicated function over the 4-sphere

I would like to maximize the following expression: $$\sum_{k = 1}^4\int_{\mathbb{S^4}}|u - \mathbf{e}_k.u_k||\nabla u_k|.\Big(1 - \frac{1}{8}\Big(\langle{v, \mathbf{e}_k}\rangle^2 + \langle{ e_k,\frac{\nabla u_k}{|\nabla u_k|}}\rangle^2 + \langle{\frac{\nabla u_k}{|\nabla u_k|}, \frac{u - u_k.\mathbf{e}_k}{u - u_k.\mathbf{e}_k}}\rangle^2 + \langle{\frac{u - u_k.\mathbf{e}_k}{u - u_k.\mathbf{e}_k}, v}\rangle^2\Big)\Big).dS$$ where $dS$ denotes the normalised measure of $\mathbb{S}^4$, $u = u_1.\mathbf{e}_1 + u_2.\mathbf{e}_2 + u_3.\mathbf{e}_3 + u_4.\mathbf{e}_4$, where $\mathbf{e}_i$ are the standard vector and $\nabla$ is the gradient on $\mathbb{S}^4$. For the maximization we ask that $|u|^2 = u_1^2 + u_2^2 + u_3^2 + u_4^2$ integrates to $1$ over $\mathbb{S}^4$ AND that $u_i$ are spherical harmonics of degree 3 over $\mathbb{S}^4$.

To do this with Mathematica, I am using a package by Sheldon Axler (https://www.axler.net/HFT_Math.html) that generates an orthonormal basis of spherical harmonics of degree 3 for me. The first step is to integrate the term for $k = 1$.

m = 4; n = 3; setDimension[x, n]; d = dimHarmonic[m, n]; basis = basisH[m, x, Sphere] /. \ {\[LeftDoubleBracketingBar]x\[RightDoubleBracketingBar]^2 -> x^2 + y^2 + z^2 + t^2, x[1] -> x, x[2] -> y, x[3] -> z, x[4] -> t}; weight[a_?(VectorQ[#, NumericQ] &), b_?(VectorQ[#, NumericQ] &), c_?(VectorQ[#, NumericQ] &), d_?(VectorQ[#, NumericQ] &)] := 1 - 1/8*(x^2 + ({x, y, z, t} . unit[{0, b . basis, c . basis, d . basis}])^2 + (unit[spheregrad[a . basis]] . {1, 0, 0, 0})^2 + (unit[spheregrad[a . basis]] . unit[{0, b . basis, c . basis, d . basis}])^2); term[a_?(VectorQ[#, NumericQ] &), b_?(VectorQ[#, NumericQ] &), c_?(VectorQ[#, NumericQ] &), d_?(VectorQ[#, NumericQ] &)] := (1/surfaceArea[n])* NIntegrate[ Sqrt[(b . basis)^2 + (c . basis)^2 + (d . basis)^2]* norm[spheregrad[a . basis]]* weighta[a, b, c, d], {x, y, z, t} \[Element] Sphere[n], WorkingPrecision -> 15, MaxPoints -> 10000]; 

Here unit denotes the re-scaling of a vector to norm $1$, spheregrad is the function that return the spherical gradient. Evaluating the function term at a random point on $\mathbb{S}^{64}$ ($d = 16$ is the dimension of spherical harmonics of degree $3$ on $\mathbb{S}^4$, so writing $(a,b,c,d) \in \mathbb{S}^{64}$ with $u_1 = a.basis$, $u_2 = b.basis$ and so on) simply gives Warnings `` and the function returns no answer. I suspect the reason is that the function weight is complicated. Any help in making term return some value is much appreciated!!