Nested square brackets in tensor indices

Question

I know that using square brackets on tensor indicies denote the anti-symmetric part

$$ T_{[ab]} = \frac{1}{2} \left( T_{ab} - T_{ba} \right)$$

I now have to prove that

$$ T_{a [[bc]d]} = T_{a [bcd]} $$

But I'm not sure what to do with the nested square brackets. Does this mean the following?

$$ T_{a [[bc]d]} = \frac{1}{2} \left( T_{a[bc]d} - T_{ad[bc]} \right)$$

Answer 1

No, the expansion of last expression on the right does not have terms with $d$ in the third slot, whereas $T_{a[bcd]}$ does.

One doesn't see nested brackets in practice (exactly because of the identity you're proving), but I'd interpret them to mean that one skews first over $bc$, and then skews over $bcd$. In other words, $$T_{a[[bc]d]} = \frac{1}{2}(T_{a[bcd]} - T_{a[cbd]}),$$ and at this stage the proof should be easy.