Sorry for the initial mistake. $\tau\lambda^a\mu^b\lambda^c\mu^d=0$ should read $\tau_{abcd}\lambda^a\mu^b\lambda^c\mu^d=0$. However, my approach to this problem is to introduce vectors, $\alpha$ and $\beta$ such that, I could use them for expansion. i.e. $\tau_{abcd}\lambda^a(\alpha^b+\beta^b)\lambda^c(\alpha^d+\beta^d)$. How to get out from there is my problem.\
A type $(0,4)$ tensor $\tau_{abcd}$ satisfies $\tau \lambda^a\mu^b\lambda^c\mu^d=0$ for all contravariant vectors $\lambda^a$ and $\mu^b$. Show that its components satisfy $\tau_{abcd}+\tau_{cbad}+\tau_{adcb}+\tau_{cdab}=0$