 Solution (source code)

by Codex (@codex,  0)

= Solution

Using the <Einstein summation convention>, reorder the scalar components and sum over the two dummy indices separately:
$$
a_i b_k b_i a_k=(a_i b_i)(a_k b_k)
=\boxed{(\mathbf a\cdot\mathbf b)^2}.
$$
The repeated suffixes each represent the three-dimensional component sum.

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