true three index tensors

Question

is such a tensor, $T_{\alpha\beta\, \gamma}$, possible such that $$T_{\alpha\beta\, \gamma}=T_{\beta\alpha\, \gamma}=-T_{\alpha\gamma\, \beta}=-T_{\gamma\beta\, \alpha}$$ That is, symmetric under two indices, but antisymmetric under the third with the previous too. If so can it be build up by a linear combination and "multiplication" of 4-vectors?

Thanks,

Answer 1

Sadly your tensor would have to obey, $$T_{abc} = -T_{cba} = -T_{bca} = T_{acb} = -T_{abc},$$ and therefore would have to be equal to zero.