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#1: Lagrange's formula

Formula: \mathbf{(a \times b )\times c} = \sum_{i,j,k=1}^3 \varepsilon_{ijk} \mathbf{(a \times b})_ic_j\mathbf{e_k} =\sum_{i,j,k=1}^3 \varepsilon_{ijk} \left(\sum_{l,m=1}^3 \varepsilon_{ilm} a_l b_m\right) c_j\mathbf{e_k} = \sum_{i,j,k,l,m=1}^3\varepsilon_{ijk}\varepsilon_{ilm} a_l b_m c_j\mathbf{e_k} = \sum_{j,k,l,m=1}^3 \left(\delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}\right) a_l b_m c_j\mathbf{e_k} =\sum_{j,k=1}^3 \left(a_j b_k c_j - a_k b_j c_j \right)\mathbf{e_k} =\mathbf {(ac)b} - \mathbf {(bc)a}


#2: contracted epsilon identity

Formula: \sum_{i=1}^3 \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}



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