Yeah the one lever isn't connected to anything but I like having the hoods and wanted symmetry I also had the shop set it up moto style so the right lever controls the front brake Only been riding fixed for a couple of years and working up to maybe eventually riding brakeless

I also have both brake hoods on my fixed gear but no lever on the non-operational one... which is why your set-up for a moment puzzled me. Which reminds me... there's a Dutch woman locally who cycles and her bike has no brakes. With traditional Dutch bikes one pedals backwards to slow down and stop!

I saw quite a few brake-less fixies when I visited my three daughters and their offspring the USA and Canada in 2018. Of course it's not legal in Europe and I've even been questioned in France by local gendarmes why I only have 'un frein'. However, it's a wonderfully free way to cycle without gears!