In many matrix applications it is possible to utilize structure for improved computational efficiency and obtain a low-rank approximation which preserves the original structure. We seek to translate efficient, low-rank, structure preserving properties to tensors, or high dimensional matrices. After unfolding a tensor into a matrix, tensor computations turn into matrix computations. When the original tensor is structured, the matrix unfolding is also structured. Motivated by a problem in computational quantum chemistry which involves a four level nested summation over a low-rank symmetric 4-tensor, we obtain a structured matrix unfolding and apply StructLDL: a rank-revealing lazy-evaluation symmetric-pivoting $LDL^T$ factorization algorithm which preserves block symmetry, symmetric blocks, and perfect shuffle permutation symmetry. StructLDL is implemented in MATLAB and allows us to compute the quantum chemistry summation in just $O(r^2n^2)$ where $r$ is the rank of the symmetric tensor. This is joint work with Charles Van Loan.