 Solution

After the reduction in part (b), multiply by $S$. The new bottom-left entry is $r$, whose absolute value is strictly smaller than the previous $|c|$:Repeated Euclidean division must terminate because the nonnegative integers $|c|$ decrease strictly until zero. Thus a word $W$ in $S,T$ and their inverses gives a matrix with bottom-left entry zero. Its determinant is one, so the diagonal entries are equal and are either both $1$ or both $-1$:Since $S^2=-I$, this final matrix belongs to $⟨S,T⟩$. So does $W^{-1}$, proving