SNI Implementation

Thank you both for the explanations. I should have added in my original post that this was my first time working with anything SNI, so maybe I have a misunderstanding that I haven't realized yet.

I now understand that not seeing the server_name extension from the server side of the BIGIP is the expected behavior, but that design decision confuses me. Currently, without the BIGIP involved, the web front end receives client hello packets with this extension and it works properly. So why wouldn't the BIGIP be designed to send them?

On the web front end, IIS currently has only two sites. One is configured to require SNI (this is a check box in the binding section), the other site has the box unchecked. The box being checked for the one site gave me the impression that the site won't work without the server_name extension.

In the mean time, I will attempt my configuration like the diagram by Rodrigo.