Symmetric and antisymmetric parts

Given a tensor $T_{i j k}$ (with three indices as an example), the symmetric and antisymmetric parts of the tensor with respect to a pair of indices (let's say $i$ and $j$ ) can be defined as follows:

Symmetric Part:

The symmetric part of the tensor $T_{i j k}$ with respect to the indices $i$ and $j$ is defined as:

T_{(i j) k} = \frac{1}{2} (T_{i j k} + T_{j i k})

Antisymmetric Part:

The antisymmetric part of the tensor $T_{i j k}$ with respect to the indices $i$ and $j$ is defined as:

T_{[i j] k} = \frac{1}{2} (T_{i j k} - T_{j i k})

Generalization to More Indices:

If the tensor has more indices, you can symmetrize or antisymmetrize it with respect to any pair (or set) of indices. For example, if $T_{i j k l}$ is a fourth-rank tensor, you can find the symmetric and antisymmetric parts with respect to the indices $i$ and $j$ as:

Symmetric part with respect to $i$ and $j$ :

T_{(i j) k l} = \frac{1}{2} (T_{i j k l} + T_{j i k l})

Antisymmetric part with respect to $i$ and $j$ :

T_{[i j] k l} = \frac{1}{2} (T_{i j k l} - T_{j i k l})