This image actually came from older news (Screenrant), back at the beginning of this year, when it was debated whether the State of the Union address broadcast was going to co-opt the premiere of the final season of Lost.
It was a showdown indeed. And on January 8, White House Press Secretary Robert Gibbs backed down to Lost.
Damon and Carlton had been holding their breath, and were jubilant about owning Groundhog day. See Darlton's Twitter here.
I leave you with one final Barack Obama DHARMA crossover, from an article from EW.
Product sponsorship from Lost as recently seen during a recent television commercial break in a nearby alternate universe.
These are both from this Lost caption thread at Lost-forum.com.
The animated gif at top is based on a post on the third page of the thread from user "nightskiesfading"-- The images he/she used are no longer hosted, but from the filenames from the text-only archive I deduced they were numbered images taken from lost-media.com, so I found them and put them together into a (yes very random) animated gif just for you. The bottom image is from the first page of the thread from user "nitromoose".
Please check out the thread for many more!