Cartan integers are preserved by isomorphism? Suppose you have two roots systems $\Phi\subset E$ and $\Phi'\subset E'$ with bases of simple roots $(\alpha_1,\dots,\alpha_\ell)$ and $(\alpha_1',\dots,\alpha_\ell')$ such that the Cartan integers $\langle\alpha_i,\alpha_j\rangle=\langle\alpha_i',\alpha_j'\rangle$ for all $i,j$, then the isomorphism $\phi$ defined by $\alpha_i\mapsto\alpha_i'$ preserves Cartan integers for any roots. That is, if $\alpha,\beta\in\Phi$, then $\langle\alpha,\beta\rangle=\langle\phi(\alpha),\phi(\beta)\rangle$. 
Every source I read says this follows easily from the formula for a reflection, looking at it, if $\sigma_{\phi(\alpha)}$ is the reflection about the root $\phi(\alpha)$, 
$$
\sigma_{\phi(\alpha)}(\phi(\beta))=\phi(\beta)-\langle\phi(\beta),\phi(\alpha)\rangle\phi(\alpha)
$$
but what more can you do to see $\langle\alpha,\beta\rangle=\langle\phi(\alpha),\phi(\beta)\rangle$?
 A: First, let $\beta$ be any root, and $\alpha$ be a simple root in $\Phi$. Writing $\beta=\sum_{i=1}^\ell c_i\alpha_i$, note that
$$
\langle\phi(\beta),\phi(\alpha)\rangle=\sum_{i=1}^\ell c_i\langle\phi(\alpha_i),\phi(\alpha)\rangle=\sum_{i=1}^\ell c_i\langle\alpha_i,\alpha\rangle=\langle\beta,\alpha\rangle
$$
where we've used the fact that $\langle\cdot,\cdot\rangle$ is at least linear in the first coordinate, and the second equality follows by hypothesis.
Now if $\alpha$ is any root, $\alpha=\sigma\alpha_i$ for some simple root $\alpha_i$ and $\sigma\in W$, since every root is conjugate to a simple root under the action of the Weyl group $W$. So 
$$
\phi(\alpha)=\phi\sigma\alpha_i=(\phi\sigma\phi^{-1})(\phi\alpha_i)
$$
and $\phi\sigma\phi^{-1}\in W'$, the Weyl group of $E'$. Then
$$
\begin{align*}
\langle\phi(\beta),\phi(\alpha)\rangle &=\langle \phi(\beta),\phi\sigma\phi^{-1}\phi\alpha_i\rangle\\
&= \langle\phi\sigma^{-1}\beta,\phi\alpha_i\rangle\\
&=\langle \sigma^{-1}\beta,\alpha_i\rangle\\
&=\langle\beta,\sigma\alpha_i\rangle\\
&= \langle\beta,\alpha\rangle.
\end{align*}
$$
Here the third equality follows from the discussion above, and the second and fourth equalities follow from applying the inverses of the elements of the Weyl group to both coordinates, which doesn't change the value.
