No it is not. You abused the Einstein summation convention. An index is supposed to appear only once as an upper index and only once as a down index. Here in the contraction of the metric with itself, you have to rename $\mu \to \alpha$ and $\nu \to \beta$ and then you will be able to see your mistake.

You can find more on the subject here.

No it is not. You abused the Einstein summation convention. An index is supposed to appear only once as an upper index and only once as a lower index. Here in the contraction of the metric with itself, you have to rename $\mu \to \alpha$ and $\nu \to \beta$ and then you will be able to see your mistake.

You can find more on the subject here.