Referring to the answer in the following question:
https://physics.stackexchange.com/a/349030/288587
I just cant figure out how to go from:
$$ \eta_{\mu\nu} = \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\nu}\eta_{\alpha\beta} $$
to:
$$ η=Λ^TηΛ $$
I tried to do the following:
$$ \begin{align} \eta_{\mu\nu} &= \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\nu}\eta_{\alpha\beta} \\ &= \Lambda^\alpha_{\;\mu}\Lambda_{\alpha\nu} \\ &= \Lambda^\alpha_{\;\mu}\eta_{\alpha\sigma}\Lambda^\sigma_{\;\nu} \\ &= \Lambda^\nu_{\;\mu}\eta_{\nu\mu}\Lambda^\mu_{\;\nu} \end{align} $$
Implying that the transpose of $\Lambda^\mu_{\;\nu}$ is $\Lambda^\nu_{\;\mu}$ but, as i know, this is not true, as the transpose of $\Lambda^\mu_{\;\nu}$ should be $\Lambda_{\nu\;}^{\;\mu}$.
Where have i gone wrong?
