A simple question on root systems

I read the Wikipedia page about root systems. It is very easy to verify that the 2 following images are showing some root systems.

https://en.wikipedia.org/wiki/Root_system

enter image description here

These 2 pictures are from Wikipedia. I think if we rotate one of them $10^{\circ}$, for example if we rotate the 1st, or rotate it $\alpha^{\circ}$, then again we have a root system, because all of the four conditions stated there remain invariant.

But why we just consider only $45^{\circ}$ rotation?

I think there is something very simple, which I can not see, but I can not find it.

edit: In comments, it is mentioned that one is related to $A_1\times A_1$, and the other one is related to $D_2$. My question is why we can not consider $10^{\circ}$ rotation? Why it is not working?

edit: I think if we rotate the root system $A_1\times A_1$ exactly $45^{\circ}$ then we have the root system of $D_2$. Am I mistaken? What do the diagrams that differ by rotation (one is obtained by the rotation of the other one) have to do with each other? What is the difference between the root system of $A_1\times A_1$ and its $10^{\circ}$-rotation? What is the difference between the root system of $A_1\times A_1$ and its $45^{\circ}$-rotation (which I think is $D_2$)?

edit: Why do we just consider the system root of $A_1\times A_1$, and its $45^{\circ}$-rotation? Why we do not consider the system root of $A_1\times A_1$, and its $10^{\circ}$-rotation, and its $20^{\circ}$-rotation? What is special about $0^{\circ}$-rotation of the system root of $A_1\times A_1$, and $45^{\circ}$-rotation of the system root of $A_1\times A_1$?

Solution 1:

The two important features of a root system are the angles between, and the relative lengths. Rotating the whole thing does not change either.

From my viewpoint, the $D_2$ root system is literally the same as (meaning isomorphic to) the $A_1\times A_1$. Indeed, the corresponding Lie algebras are isomorphic.

It's true that usually the simple roots are aligned in traditional ways, but that is mathematically inessential.

Continuing Solutions of $\dot{x} = \frac{t^2x^5}{1 + x^2 + x^4}$ to entire number line

Why are the order statistics of uniform Beta?

Should a polynomial satisfying certain conditions be linear?

Eigenvalue of self-adjoint

If $\varphi, g$ are positive and non-zero in $L^p(\mathbb R^n)$ and $L^q(\mathbb R^n)$ respectively, then so is their product in $L^1(\mathbb R^n)$

Find all complex numbers $z$ such that $| z | = 1$ and $ |\frac z {\overline z} + \frac {\bar z} z | = 1. $ [closed]

I managed to solve this uniform distribution problem and now i am having difilculties with calculating area on graph

If a changes to b, then doesn't a + d = b? Why a(1 + d) = b? [closed]

Retrieving second order ODE from its solutions

Find the area of region ABC below

Finite normal subgroups of $\operatorname{SU}(n)$