I have a following problem:
"Calculate the average values of the products of the components of the unit vectors: $$\langle n_i \rangle, \langle n_i n_j \rangle, \langle n_i n_j n_k \rangle, \langle n_i n_j n_k n_l \rangle, \langle n_i n_j n_k n_l n_m \rangle$$ Averaging is performed over a circle perpendicular to the unit vector $\vec{h}: h_i h_i = 1$."
I have a solution to this problem, but I can't fully understand it.
First, it is easy to understand that $$\langle n_i \rangle = \langle n_i n_j n_k \rangle = \langle n_i n_j n_k n_l n_m \rangle = 0$$
Therefore, we consider only $\langle n_i n_j \rangle$ and $\langle n_i n_j n_k n_l \rangle$
The solution says that $\langle n_i n_j \rangle$ can be searched in the form: $$\langle n_i n_j \rangle = C \delta_{ij} + D h_i h_j$$ And $\langle n_i n_j n_k n_l \rangle$ can be searched in the form: $$\langle n_i n_j n_k n_l \rangle = C_1 (\delta_{ij} \delta_{kl}+ \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}) + C_2 h_i h_j h_k h_l + C_3 (\delta_{ij} h_k h_l + \delta_{ik} h_j h_l + \delta_{il} h_j h_k + \delta_{jk} h_i h_l + \delta_{jl} h_i h_k + $$ $$ + \delta_{kl} h_i h_j)$$ The coefficients in these expressions are not difficult to find, but I do not understand why the answer has such a form? How to prove it?
