How would you use index notation to show that these vector equations are equal?

Question

How would you use index notation to show that $(\underline{a} \times \underline{b}) \cdot (\underline{a} \times \underline{b}) = |\underline{a}|^{2} |\underline{b}|^{2}-(\underline{a} \cdot \underline{b})^{2}$

My attempt to do this was as follows:

$(\underline{a} \times \underline{b}) \cdot (\underline{a} \times \underline{b}) = \varepsilon_{ijk}a_{j}b_{k} \varepsilon_{ijk}a_{j}b_{k} \\ = \varepsilon_{ijk}\varepsilon_{ijk}a_{j}a_{j}b_{k}b_{k} \\ = 6|\underline{a}|^{2} |\underline{b}|^{2}$

What am I doing wrong here?

Answer 1

The first line should be

$(\underline{a} \times \underline{b}) \cdot (\underline{a} \times \underline{b}) = \varepsilon_{ijk}a_{j}b_{k} \varepsilon_{imn}a_{m}b_{n}$.

Reason: The two cross products are independent, so you should use different indices for the summation.