Contravariant Tensors 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 
$$\tau_{abcd}+\tau_{cbad}+\tau_{adcb}+\tau_{cdab}=0$$
 A: We have for every $\lambda$, $\xi$, $\mu$:
\begin{align*}
  0 &= \tau_{abcd}(\lambda^a + \xi^a)\mu^b(\lambda^c + \xi^c)\mu^d\\
    &= \tau_{abcd}\lambda^a\mu^b\lambda^c\mu^d + \tau_{abcd}\xi^a\mu^b\lambda^c\mu^d + \tau_{abcd}\lambda^a\mu^b\xi^c\mu^d + \tau_{abcd}\xi^a\mu^b\xi^c\mu^d\\
    &= \tau_{abcd}\xi^a\mu^b\lambda^c\mu^d + \tau_{abcd}\lambda^a\mu^b\xi^c\mu^d\\
    &= (\tau_{abcd} + \tau_{cbad})\xi^a\mu^b\lambda^c\mu^d
\end{align*}
and hence, for every $\lambda$, $\xi$, $\mu$, $\nu$:
\begin{align*}
   0 &= 
   (\tau_{abcd} + \tau_{cbad})\xi^a(\mu+\nu)^b\lambda^c(\mu+\nu)^d\\
   &= (\tau_{abcd} + \tau_{cbad})\xi^a\mu^b\lambda^c\mu^d+
   (\tau_{abcd} + \tau_{cbad})\xi^a\nu^b\lambda^c\mu^d + 
   (\tau_{abcd} + \tau_{cbad})\xi^a\mu^b\lambda^c\nu^d
   + (\tau_{abcd} + \tau_{cbad})\xi^a\nu^b\lambda^c\nu^d\\
  &= (\tau_{abcd} + \tau_{cbad})\xi^a\nu^b\lambda^c\mu^d + 
   (\tau_{abcd} + \tau_{cbad})\xi^a\mu^b\lambda^c\nu^d\\
  &= (\tau_{abcd} + \tau_{cbad} + \tau_{adcb} + \tau_{cdab})\xi^a\mu^b\lambda^c\nu^d
\end{align*}
As $\xi$, $\mu$, $\nu$ and $\lambda$ were arbitrary, is follows 
\[ 
  0 = \tau_{abcd} + \tau_{cbad} + \tau_{adcb} + \tau_{cdab}.
\]
