Using Gmail smtp server from within Gmail

I have a gmail account. In my new job, the institutional e-mail is a customized gmail e-mail account. Trying to simplify things, I am forwarding the institutional e-mail do my gmail account. That was simple.

The next step was to make the generic gmail account to compose e-mails both as standard gmail and the institutional one. For that I can choose to use the gmail smtp server or a custom one. When using the gmail smtp server, e-mails have a From field in the header with the institutional e-mail, and a Sender field with the generic gmail e-mail. That is not elegant.

I was trying to configure gmail with a custom smtp server (the institutional smtp server, that is gmail smtp server), in the hope that hides the Sender field. Unfortunately I am without luck. It doesn’t work…