= Solution
The integration domain is invariant under every <rotation matrix>, and its volume element is unchanged. The substitution $\mathbf y=R\mathbf x$ therefore proves
$$
R_{ia}R_{jb}R_{kc}R_{ld}I_{abcd}=\int_B y_iy_jy_ky_l\,dV_{\mathbf y}=I_{ijkl}.
$$
Thus $I$ is an <isotropic tensor>. It is also completely symmetric in its four indices, because the coordinate factors commute. In the general isotropic form, exchanging indices $j,k$ equates $\alpha$ and $\beta$, and exchanging $j,l$ equates $\alpha$ and $\gamma$. Therefore
$$
\boxed{I_{ijkl}=A(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),\qquad\alpha=\beta=\gamma=A.}
$$
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