After discussion in the comments, I understand the question as follows: Say we have started with the Lie algebra $\mathfrak{sl}_3(K) = \lbrace A \in M_{3\times 3}(K): tr(A)=0 \rbrace$ and have chosen as the most obvious Cartan subalgebra the one consisting of the diagonal matrices in $\mathfrak{sl}_3(K)$ i.e.
$\mathfrak{h} = \lbrace \pmatrix{a&0&0\\0&b&0\\0&0&c}: a+b+c=0 \rbrace $,
as well as the roots $\pm\alpha, \pm \beta, \pm \gamma \in \mathfrak{h}^*$ given by
$\alpha(\pmatrix{a&0&0\\0&b&0\\0&0&c})= a-b$,
$\beta(\pmatrix{a&0&0\\0&b&0\\0&0&c})= b-c$,
$\gamma(\pmatrix{a&0&0\\0&b&0\\0&0&c})= a-c$.
We might also have found the corresponding root spaces $\mathfrak{g}_\alpha = \pmatrix{0&*&0\\0&0&0\\0&0&0}$ to $\alpha$, $\mathfrak{g}_{-\alpha} = \pmatrix{0&0&0\\*&0&0\\0&0&0}$ to $-\alpha$, $\mathfrak{g}_{\alpha+\beta} =\pmatrix{0&0&*\\0&0&0\\0&0&0}$ to $\alpha+\beta$ etc. (We can call $\pmatrix{0&1&0\\0&0&0\\0&0&0}$ a "root vector" to the root $\alpha$, but so is $\pmatrix{0&17&0\\0&0&0\\0&0&0}$.)
Now the question is: How do we get, from this, that the root system looks like the picture in the OP?
A shortcut is to to notice that $\gamma=\alpha+\beta$ and to use the classification of root systems, which tells us that the only root system which consists of six roots, three of which positive, one of which is the sum of the other two, is the root system $A_2$, and ten thousand people before us have checked that that root system looks as the picture in the OP.
The more rewarding way is as follows: A full description of a root system and its "geometry" needs the coroots $\check{\alpha}, \check{\beta} ...$ to the roots. Namely, the crucial relations are
$s_{\alpha}(\beta) = \beta-\check{\alpha}(\beta) \beta$ (reflection of $\beta$ at the hyperplane perpendicular to $\alpha$), and more or less equivalently
$(\ast) \qquad \check{\alpha}(\beta) \cdot \check{\beta}(\alpha) = 4 \cos^2(\theta)$ where $\theta$ is the angle between $\alpha$ and $\beta$.
With these formulae we get the "realisation" of our root system in an Euclidean space, as soon as we have those coroots. One obvious property we need is $\check{\rho}(\rho) = 2$ for all roots, but where are those coroots in the Lie algebra? They are realised as special elements of $\mathfrak{h}^{**} \simeq \mathfrak{h}$, as follows: For each root $\rho$, the space $[\mathfrak{g}_\rho, \mathfrak{g}_{-\rho}]$ is a one-dimensional subspace of $\mathfrak{h}$, and it contains a
unique element $H_{\rho} \in [\mathfrak{g}_\rho, \mathfrak{g}_{-\rho}]$ such that $\rho(H_\rho) =2$.
This element $H_\rho \in \mathfrak{h}$ is the coroot $\check{\rho}$ via the identification $\mathfrak{h}^{**} \simeq \mathfrak{h}$. (Down to earth: For each root $\sigma$, $\check{\rho}(\sigma) = \sigma(H_{\rho})$.)
In our example, we get
$[\mathfrak{g}_\alpha, \mathfrak{g}_{-\alpha}] = \lbrace \pmatrix{a&0&0\\0&-a&0\\0&0&0} : a \in K \rbrace$ and hence $H_\alpha = \pmatrix{1&0&0\\0&-1&0\\0&0&0}$, and likewise
$H_\beta= \pmatrix{0&0&0\\0&1&0\\0&0&-1}$, $H_\gamma= \pmatrix{1&0&0\\0&0&0\\0&0&-1}$, $H_{-\alpha} = \pmatrix{-1&0&0\\0&1&0\\0&0&0}$ etc.
In particular, $\check{\alpha}(\beta) = \beta(H_\alpha) = -1$ as well as $\check{\beta}(\alpha) = \alpha(H_\beta)=-1$ which together with $(\ast)$ tells us that the angle between $\alpha$ and $\beta$ has cosine $1/2$ or $-1/2$, i.e. is either $60°$ or $120°$. Checking the other combinations quickly shows that it must actually be $120°$, that $\alpha+\beta$ sits exactly "between" $\alpha$ and $\beta$ with and angle of $60°$ to each, and consequently that the root system looks as in the picture.
Added in response to comment: Be aware that $\rho(H_\rho) = 2$ is true by definition for all $\rho$, regardless of the length of $\rho$. Rather, the ratio of two root lengths is given by
$$\dfrac{\lvert \lvert \beta \rvert \rvert^2}{\lvert \lvert \alpha\rvert \rvert^2} = \dfrac{\beta(H_\alpha)}{\alpha(H_\beta)}$$
(and, as is well known, can only take the values $3,2,1,\frac12, \frac13$).
[In many sources they write something like $\langle \beta, \check{\alpha} \rangle$ for $\beta(H_\alpha)= \check{\alpha}(\beta)$ but it is important to note that this $\langle, \rangle$ is not a scalar product (it's generally not even symmetric to begin with) and thus will not directly give us lengths. "True" scalar products are then usually denoted by something like $(\alpha \vert \beta)$, and that would give the above $\lvert \lvert \alpha \rvert \rvert^2 = (\alpha \vert \alpha)$.]
Note further that it is in general not true that $H_{\alpha+\beta} \stackrel{?}= H_\alpha + H_\beta$ (in fancier words and more precisely: the map $\rho \mapsto H_\rho$ is in general not a morphism, let alone isomorphism, of root systems $\Phi \rightarrow \check{\Phi}$). To see a concrete example, look at a form of type $B_2$, i.e. (compare https://math.stackexchange.com/a/3629615/96384, where the matrices are mirrored at the diagonal, and the notation for $\alpha$ and $\beta$ is switched)
$\mathfrak{so}_5(\mathbb C) := \lbrace \pmatrix{a&b&0&e&g\\
c&d&-e&0&h\\
0&f&-a&-c&i\\
-f&0&-b&-d&j\\
-i&-j&-g&-h&0\\} : a, ..., j \in \mathbb C \rbrace$.
Exercise: There is a long root $\beta$ and a short root $\alpha$ such that a system of simple roots is given by $\beta, \beta+\alpha, \beta+2\alpha, \alpha$, and we have
$$H_\beta = \pmatrix{1&0&0&0&0\\
0&-1&0&0&0\\
0&0&-1&0&0\\
0&0&0&1&0\\
0&0&0&0&0\\}, H_\alpha= \pmatrix{0&0&0&0&0\\
0&2&0&0&0\\
0&0&0&0&0\\
0&0&0&-2&0\\
0&0&0&0&0\\},$$
but
$$H_{\alpha+\beta} = \pmatrix{2&0&0&0&0\\
0&0&0&0&0\\
0&0&-2&0&0\\
0&0&0&0&0\\
0&0&0&0&0\\} \neq H_\alpha +H_\beta.$$
Note that as said $\rho(H_\rho)=2$ for all roots, so we cannot tell long from short roots with that; but $\beta(H_\alpha) = -2, \alpha(H_\beta)=-1$ which makes $\beta$ longer than $\alpha$ by a factor of $\sqrt2$, and with the method from above, we get that the angle between $\alpha$ and $\beta$ is $3\pi/4 \hat{=} 135°$, and the root system looks like this: https://commons.wikimedia.org/wiki/File:Root_system_B2.svg
Best Answer
As far as I understand, it's as follows: to extend the scalars of a Lie group or Lie algebra is pretty easy. Well OK, I remember when I was a beginner tensor products freaked me out, and to be honest base extensions of Lie/algebraic groups still do sometimes, but in spirit it's a pretty straightforward concept. You allow more coefficients. It's as if you've come to like some person, and now you see them dressed up in fancy clothes and with some jewellery; that can be nice and interesting, but underneath you know it's still the person you got to know and like when they wore a shirt and jeans.
But the backward process is hard. (There is a very easy thing called restriction of scalars, but in a way this is not the "right" reversal of that scalar extension that we are looking for.) Imagine we are given some Lie group or algebra $S$ over $\mathbb C$, all fancy with imaginary units and stuff. But is it already defined over $\mathbb R$ -- meaning very roughly, is there some object $R$ defined entirely with plain old real numbers, such that $S$ is just the scalar extension (dressed up version) of $R$? This is in general a very hard question. If you've only met some person in a fancy dress, can you imagine them in streetclothes?
But to find that such a group or algebra is actually defined over $\mathbb Z$ is like to be able to see that person without any clothes.
To steer away from a not necessarily too good metaphor to something more mathematical, the ring $\mathbb Z$ is the basis of all rings (technically, it is the initial object in the category of rings), meaning that if you have something that is defined over $\mathbb Z$, then from there you can go "up" again to any other ring. (One last flashback to my metaphor: Now you can dress them up with any clothes you like.)
For example, say you want to study Lie-type groups over finite fields. To do a base change between two fields, you need a homomorphism between those fields. More generally for changing scalars from some ring $R$ to some other ring $S$, you need some homomorphism $R \rightarrow S$; which, for fields, will always be plain inclusions $K \subset L$.
So it's easy to change scalars from $\mathbb R$ to $\mathbb C$. Or from $\mathbb C$ to $\mathbb C(t)$. Or from $\mathbb Q$ to $\mathbb R$. But not the other way around. And from none of them is there any chance to get to $\mathbb F_p$.
But now imagine you started with something over $\mathbb C$ or $\mathbb R$ and you have managed to strip it down to something over $\mathbb Z$. Or Chevalley did it for you. Well now you have a map $\mathbb Z \rightarrow \mathbb F_p$, so now you can change its base to your favourite finite field! As far as I understand, that was a big thing which was now allowed through Chevalley's $\mathbb Z$-basis, e.g. Steinberg then constructed analogues of some classically "real" stuff over finite fields, giving some cool new constructions in the theory of finite groups.
And even if you're not interested in finite groups and all those base change possibilities, still having something defined over $\mathbb Z$ somehow feels as if you can see it clearly, with all clutter of $\sqrt2$'s and $\pi$'s and $i$'s removed.