# double symmetrization in Eistein index notation (Physics SE duplicate)

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What does the following supposed to give? This appears in equation (3.12) in the following link: https://arxiv.org/pdf/1312.5344.pdf (last term)

$$A^{(I(J}B^{K)L)} =$$

$$1) \quad \frac{1}{2}(A^{(IJ}B^{KL)}+A^{(IK}B^{JL)})=\frac{1}{4}(A^{IJ}B^{KL}+A^{KL}B^{IJ}+A^{IK}B^{JL}+A^{JL}B^{IK})$$
$$2) \quad \frac{1}{2}(A^{(IJ}B^{KL)}+A^{(IK}B^{JL)})=\frac{1}{4}(A^{IJ}B^{KL}+A^{LJ}B^{KI}+A^{IK}B^{JL}+A^{LK}B^{JI})$$

$$\text{3)} \quad \text{Something else?}$$

Here, I assume that the author is using $A^{(I}B^{J)}=\frac{1}{2}(A^I B^J+A^J B^I).$

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