the normal subgroup of $SL(3,\mathbb{Z})$

Question

I know we can use Margulis normal subgroup theorem to prove the normal subgroup of the finite index subgroup of $SL(3,\mathbb{Z})$ is either finite or finite index, could we prove this fact without Margulis normal subgroup theorem? Since Margulis normal subgroup theorem states the situation more general, but now we just focus on $SL(3,\mathbb{Z})$.