Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 5A / a / i / Solution

Both sides are antisymmetric in $j,k$ and in $p,q$. If either pair has equal indices, both sides vanish. It therefore suffices to take $j\ne k$ and $p\ne q$. The sum over $i$ is nonzero exactly when $(p,q)=(j,k)$ or $(k,j)$; it is respectively $1$ or $-1$. This is exactlythe contraction of two Levi-Civita symbols.