Recently I am reading the Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.

Let $X$ be an unimodular integral lattice, and $H$ denotes the unimodular lattice which has a basis $x$, $y$ satisfying $x\cdot x=y\cdot y=0$, and $x\cdot y=1$. For $\omega \in X$ with $\omega\cdot\omega\in 2\mathbb{Z}$, we define the isometry $E^1_\omega$ of $X\oplus H$ as follows.

• For $\xi\in X$, $E^1_\omega(\xi)=\xi-(\xi\cdot\omega)x$.
• $E^1_\omega(x)=x$.
• $E^1_\omega(y)=y+\omega-2^{-1}(\omega\cdot\omega)x$.

My question is: What is intuition behind the definition of $E^1_\omega$?

Thank you for your help.