I am currently finishing up a PhD in Statistics in the United States and am currently applying to postdocs as the next phase of my academic career. I also did my undergraduate studies in Statistics from one of the top 5 programs and am currently attending a slightly less prestigious PhD program (say top 25). As I've been looking for postdoc positions, I saw that my undergraduate adviser (who supervised my honors thesis) had posted that they were looking for a postdoc, and it seems like the type of research that I'm really interested in and is closely aligned with my current research, so I'd like to apply for this position. However, I don't know the proper protocol to apply for a position with a professor who already had a working relationship with you as your adviser (and they also had supported my graduate applications when I was applying to graduate school).
For more context, I have been keeping up with this professor, but not extremely closely (we send each other a few emails a year just to catch up). Would it be appropriate to send them an email to indicate my interest in the position (along with say a cover letter and CV?). Wording this email has been a little difficult as I'm trying to avoid sounding like I'm leveraging my connections with them to get this postdoc position. Is there a common protocol for applying to postdoc positions under professors who used to be your adviser (or more broadly a professor that you already had a working relationship in some way)? Is applying to a postdoc sponsored by a former adviser something that one should avoid doing (especially to avoid the awkwardness if you end up not getting the position)? What other considerations are there to take into account when applying to postdocs sponsored by former advisers?