 Solution

Denote a matrix by $A(x,y,z)$, with $y$ the upper-right entry. Direct matrix multiplication givesThis proves closure. Associativity is inherited from matrix multiplication. The identity is $A(0,0,0)$, and the inverse isBoth products with this matrix are the identity. Thus these real matrices form the Heisenberg group, in the coordinate convention with central coordinate $y$.

Let $N=\{A(0,b,0):b\in \mathbb{R}\}$. Its multiplication adds $b$, its identity is $b=0$ and its inverse negates $b$, so it is a subgroup. MoreoverThus every element of $N$ is central, and $gN{g}^{-1}=N$. Therefore $N$ is a normal subgroup. In fact it is the full center of a group, as described by the center quotient of the real Heisenberg group.