Why can we always choose Riemannian metrics to be symmetric?

Question

I'm reading General Relativity: An Introduction for Physicists by Hobson, Efstathiou and Lasenby, here, and I'm stuck in this part of the book (chapter 2, page 32), where it says that Riemannian metrics can always be chosen to be symmetric. I don't understand why do we interchange the indices as mentioned in the underlined phrase. Any help is appreciated.

Answer 1

For example, if $\alpha=(b+c)/2$ then $$ax^2+bxy+cyx+dy^2=ax^2+\alpha xy+\alpha yx+dy^2.$$