Chong-Yun Chao's paper "UNCOUNTABLY MANY NONISOMORPHIC NILPOTENT LIE ALGEBRAS" implies that there are uncountably many nonisomorphic nilpotent real Lie algebras (in any dimension greater than 9), but your scheme only includes countably many. I'm not sure what the simplest non example is.

@Turbo, We know that $x_1y_1+x_2y_2$ is an integer whenever $(y_1,y_2)$ is in the group generated by the four vectors. js21 shows that this group includes the vectors $(1,0)$ and $(0,1)$, which implies directly that $x_1=1\cdot x_1+0\cdot x_2$ and $x_2=0\cdot x_1+1\cdot x_2$ are integers.

Thanks Henry. It looks like Bowditch's paper answers this question on the nose.

The question is reasonably well posed, but definitely not research level. In the future, consider MathSE.

It suffices to find one choice of a_i such that S_m is linearly independent, right?

Yes, it should be $Y_{n+1}\geq t$. You can find the probability density by just taking the derivative of the cumulative density function $P(X_{n}\leq t)$ (because this function is smooth).