Jacobson radical of the ring of lower triangular

Step 1
Fix a positive integer n.
Let R be the ring of lower triangular n×n matrices over ${\displaystyle \mathbb{Z}}$, and let J be the Jacobson radical of R.
Let S be the set of elements of R which are strictly lower triangular.
It's easily verified that S is an ideal of R.
Claim: ${\displaystyle J=S}$
Proof:
Let ${\displaystyle A\in S}$
Since A is strictly lower triangular, it follows that A is nilpotent, hence ${\displaystyle I_n-A}$ is a unit of R.
Since S is an ideal such that ${\displaystyle I_n-A}$ is a unit of R for all ${\displaystyle A\in S}$, it follows that ${\displaystyle S\subseteq J}$ (Beacher proves this in the text).
Next suppose ${\displaystyle S\subset J}$ (proper inclusion).
Our goal is to derive a contradiction.
Let ${\displaystyle A\in \frac{J}{S}}$.
Since ${\displaystyle A\in J}$, it follows that ${\displaystyle I_n-A}$ is a unit of R, hence each diagonal element of ${\displaystyle I_n-A}$ is equal to either 1 or -1. But they can't all be equal to 1, else ${\displaystyle A\in S}$, hence at least one diagonal element of ${\displaystyle I_n-A}$ is equal to -1, so the corresponding diagonal element of A is equal to 2.
But ${\displaystyle A\in J}$ implies ${\displaystyle -A\in J}$, so ${\displaystyle I_n+A}$ must be a unit of R, contradiction, since ${\displaystyle I_n+A}$ has a diagonal element equal to 3.