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Выберем правый ортонормированный базис ${\displaystyle {\vec {e_{1}}},~{\vec {e_{2}}},~{\vec {e_{3}}}}$ так, чтобы

${\displaystyle {\vec {a}}=\alpha _{1}{\vec {e_{1}}}+\alpha _{2}{\vec {e_{2}}}+\alpha _{3}{\vec {e_{3}}},}$

${\displaystyle {\vec {b}}=\beta _{1}{\vec {e_{1}}}+\beta _{2}{\vec {e_{2}}},}$

${\displaystyle {\vec {c}}=\gamma _{1}{\vec {e_{1}}}.}$

Тогда

${\displaystyle \left[{\vec {b}},{\vec {c}}\right]=\left(0,0,-\beta _{2}\gamma _{1}\right),}$

${\displaystyle \left[{\vec {a}},\left[{\vec {b}},{\vec {c}}\right]\right]=\left(-\alpha _{2}\beta _{2}\gamma _{1},\alpha _{1}\beta _{2}\gamma _{1},0\right)}$

и

${\displaystyle {\vec {b}}\left({\vec {a}}\cdot {\vec {c}}\right)-{\vec {c}}\left({\vec {a}}\cdot {\vec {b}}\right)=\alpha _{1}\gamma _{1}{\vec {b}}-\left(\alpha _{1}\beta _{1}+\alpha _{2}\beta _{2}\right){\vec {c}}=\left(-\alpha _{2}\beta _{2}\gamma _{1},\alpha _{1}\beta _{2}\gamma _{1},0\right).}$

Таким образом,

${\displaystyle \left[{\vec {a}},\left[{\vec {b}},{\vec {c}}\right]\right]={\vec {b}}\left({\vec {a}}\cdot {\vec {c}}\right)-{\vec {c}}\left({\vec {a}}\cdot {\vec {b}}\right).}$