Split T,T
If you split T,T and get an ace, I would just be happy that you got a damn good hand instead of a stiff that can be busted or beaten. I wouldn't double down in that instance. Splitting 10s is one thing that will draw attention to you, and doubling on a X,A hand after splitting will likely cause a mild uproar in some places.
Also, when you split 10s and get a high value hand that most people don't hit, such as a 17-21, the dealer will most likely assume you want to stand and then just hit the other 10 with the first card. If you see it's another 10, then try to say you wanted to double your X,A hand knowing you would get a 10 for a doubled 21, you'll have an awkward situation on your hands...