ϵ_{ijk} ϵ_{k l m} = δ_{i l} δ_{jm} - δ_{im} δ_{j l}

, expressed through the Kronecker delta, gives

(a \times (b \times c))_{i} = ϵ_{ijk} a_{j} ϵ_{k l m} b_{l} c_{m} = b_{i} a_{j} c_{j} - c_{i} a_{j} b_{j} .

The last two contractions are dot products, proving the vector triple product identity

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c

b

Solution

c = M d, c_{i} = M_{ij} d_{j} .

Then the displayed tensor contraction is

(a_{i} c_{i}) b_{k} - (a_{i} b_{i}) c_{k} = (a \cdot c) b_{k} - (a \cdot b) c_{k}

, precisely the

k

a \times (b \times c)

M

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