Constant curvature for maximally symmetric spaces

I’m working through Walds GR Textbook and while reading chapter 5 I stumbled upon the question Proving constant curvature.
However, my question is how do we prove that $L$ is symmetric?
It is mentioned that $L$ is a self-adjoint map in regards of the scalar product induced by the metric $h$.

So my idea would be to try:
$<L(omega_{li} dx^l otimes dx^i), beta_{mn} dx^m otimes dx^n>=<omega_{li} dx^l otimes dx^i, L(beta_{mn} dx^m otimes dx^n)>$

But I am somewhat lost now… Probably it’s not too hard but I fail to see how to proceed