I want to show following property for the scalar product
$$ \left( \vec \omega \times \vec y,\vec \omega \times \vec y\right)=\left( \vec \omega,\vec y\times \left(\vec \omega \times \vec y\right)\right). $$
I would like to show it by using the Levi-Civita symbol $\varepsilon_{ijk}$ since I am not practised with it. My first attempt is
$$ \left(\varepsilon_{ijk}\vec e_i \omega_j y_k\right)\cdot\left(\varepsilon_{ijk}\vec e_i \omega_j y_k\right)=\omega_j\varepsilon_{ijk}\vec e_i y_k\varepsilon_{ijk}\vec e_i \omega_j y_k $$
but I do not see how I have to proceed. May you give me some adivce?
Hint: First, use different indices: $(\epsilon_{ijk}e_i\omega_j y_k) \cdot (\epsilon_{lmn}e_l\omega_m y_n)$. Then observe that $\epsilon_{ijk}\epsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$ and $e_i \cdot e_j = \delta_{ij}$.