Kůlové odvození vzorečku

\begin{aligned} (\vec{a} \times \vec{b})^{2} & = (\vec{a} \times \vec{b}) (\vec{a} \times \vec{b}) \\ = (\vec{a} \times \vec{b})_{i} (\vec{a} \times \vec{b})_{i} \\ = ε_{i j k} a_{j} b_{k} ε_{i l m} a_{l} b_{m} \\ = ε_{i j k} ε_{i l m} a_{j} b_{k} a_{l} b_{m} \\ = (δ_{j l} δ_{k m} - δ_{j m} δ_{k l}) a_{j} b_{k} a_{l} b_{m} \\ = δ_{j l} δ_{k m} a_{j} b_{k} a_{l} b_{m} - δ_{j m} δ_{k l} a_{j} b_{k} a_{l} b_{m} \\ = a_{j} b_{k} a_{j} b_{k} - a_{j} b_{k} a_{k} b_{j} \\ = a_{j} a_{j} b_{k} b_{k} - a_{j} b_{j} a_{k} b_{k} \\ = {\vec{a}}^{2} {\vec{b}}^{2} - (\vec{a} \vec{b})^{2} \end{aligned}