Split tickets for travel

I have been looking at travelling twice to NYC in short succession. I need to go for 5 days then a 7 day break then another 5 days. Clearly the 5 day costs are killing me but I have a plan and want to know if it will work.......

Fly out LHR - JFK 6th May with return ticket 23 May..(VA through UK site)

when in USA buy opposite flights - OUT JFK - LHR 11th May with return 18th May...

For this I will use my
LHR - JFK (UK) 6th May
JFK - LHR (USA) 11th May
LHR - JFK (USA) 18th May
JFK - LHR (UK) 23rd May

MUCH cheaper and also works exactly as i need it and as i never break any ticket rules i think it works...am I missing something?

This is a case of nested itineraries and was recently discussed here

It will work just fine. You can (should) buy both trips here in the UK, one will be priced in UKL and the other in USD.