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**I'm asked to prove the following using Levi-Civita/index notation:**

[itex]

(\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = [\mathbf{a,\ b, \ d}] \mathbf c - [\mathbf{a,\ b, \ c}] \mathbf d \

[/itex]

I'm able to prove it using triple product identities, but I'm completely stuck with the index notation. I was previously able to prove Lagrange's Identity with index notation, but applying similar concepts I just get stuck on the first step with the quadruple product.

Using the same first step of proving Lagrange's identity, I transformed [itex]

(\mathbf{a \times b} )[/itex] into [itex]\varepsilon_{ijk} a^j b^k[/itex] and [itex]

(\mathbf{c \times d} )[/itex] into [itex]\varepsilon_{imn} c^m d^n[/itex] but then I'm just left with [itex](\varepsilon_{ijk} a^j b^k) \mathbf{\times} (\varepsilon_{imn} c^m d^n)[/itex] which is seemingly unhelpful.

I also tried letting AxB = W and CxD = Z and transforming WxZ to index notation. Then I tried to 'un-nest' the original cross products in index notation, but it quickly ended up in a place where I couldn't understand what the different indexes represented.

Any help would be appreciated. Thanks.