Consider the identity $$B \circ A = A : \mathbb{T}: B$$
where A, B, and M are matrices and $\mathbb{T}$ is a tensor of order 6. The tensor is defined as
$$\mathbb{T}_{jklmnp} = 1$$
if $j = k = l$ and $m = n = p$. It is $0$ otherwise. Futhermore, $:$ denotes the Frobenius product and $\circ$ is the Hadamard product. I have been trying to convince myself of the validity of this expression using index notation.
What I would like clarification on is the standard way of contracting indices. When I write out the equation in index notation I get
$$(B \circ A)_{iq} = B_{ij} \mathbb{T}_{jklmnp} A_{pq}$$
Is this the correct way to contract the indices ? If so, why isn't any other contraction of the indices (say, exchanging p and q) valid?
Sorry if this is a repeat question, I tried searching before posting. Thanks in advance.