Matrix Algebra question

Let L_2 be the 2x2 lower triangular matrix whose nonzero off diagonal
entry is 2, i.e. a11=1, a12=0,
a21=2, a22=1. Let U_2 be its transpose.
I am looking for an easy proof of the following fact:

The group (with matrix multiplication) generated by {L_2, U_2} is the
set of matrices A with a11, a22 odd, and a21, a12 even, and det(A)=1.

A direct proof seems to be not easy.

-TCL
.