E8-form (Ex)
From Manifold Atlas
Let denote the negative definite even form with signature
and rank
:
![\displaystyle E_8 = \left( \begin{array}{cccccccc} -2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & -2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & -1 & -2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & -2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & -2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -2 \end{array}\right):\Z^8 \to (\Z^8)^*.](/images/math/8/3/d/83d942ee26a65b2e8954edd489739ed5.png)
Consider the forms and
. Verify that these two forms are both odd, of the same rank and signature but yet still inequivalent (non-isometric).