How should I reply to this Professor's email? The following is the last email I received from a Professor in June 2019, and recently (in Dec 2019) I submitted my Ph.D. application to this University.

"Thanks for writing and for the great ideas -- I see a lot of energy and promise in your thoughts, and I encourage you to try to pursue some of these questions -- most, in fact, are exciting and open research areas. I also encourage you to apply for PhD at YYY University.

Best wishes,
Professor X" 

Now that I have submitted my application, I want to send a simple email to this Professor reminding her about my interest to join her group and the discussion we had 4 months back (through emails).   

Can anyone suggest a draft of the email that I can send to this Professor?
Can anyone suggest a draft of the email that I can send to this Professor? 


Reply to the Professor's original email, so that your response appears in the same thread; perhaps include the original snippet, i.e., 


  Thanks for writing and for the great ideas -- I see a lot of energy and promise in your thoughts, and I encourage you to try to pursue some of these questions -- most, in fact, are exciting and open research areas. I also encourage you to apply for PhD at YYY University.


and write something along the lines of: 

Further to your positive encouragement, I have submitted my PhD application to YYY University.

Thereafter, it depends on your agenda. E.g., you might want to reopen discussion of  joining her group, for which you could use:

Given the seeming alignment between our research interests, I would like to join your group. Can we discuss that possibility, perhaps by telephone?