(Cross-posted from math.SE since I'm not sure what platform is suitable -- see https://math.stackexchange.com/questions/3331104/root-lattices-and-resolutions-of-singular-cubic-surfaces)

Given a smooth cubic surface $X$ (say over $\mathbb{C}$) considered as a blowup of $\mathbb{P}^2$ at $6$ points) with Neron-Severi group $\operatorname{NS}(X)$ and canonical divisor $K_X$, the subset $R = \{ \alpha \in \operatorname{NS}(X) : \alpha\cdot K_X = 0, \alpha\cdot\alpha = -2\}$ is the root system of the Weyl group of $E_6$. I understand that there are previous questions on MSE related to this lattice (e.g. https://math.stackexchange.com/questions/1477220/27-lines-on-a-cubic-surface and https://math.stackexchange.com/questions/82199/automorphism-group-of-the-configuration-of-lines-on-a-cubic-surface-and-quadrati), but I haven't been able to find an explanation about what happens for *singular* cubic surfaces (on this site or other sources).

More specifically, suppose that $\varphi : X' \longrightarrow X$ is a "minimal" resolution of singularities of a singular cubic surface $X$ with rational double points as singularities. Suppose that $X \subset \mathbb{P}^3$ has rational double points as singularities. For simplicity, we take $X$ to have only one singular point (which we take to be $(1 : 0: 0 : 0)$) with singularity type $A_1$ or $A_2$. Then, what sublattice of the root system described above for $X'$ does the components of the exceptional divisor of this resolution generate?

Under the assumptions above, we can write $X = (f = 0)$ with $f(t_0, t_1, t_2, t_3) = t_0 g_2(t_1, t_2, t_3) + g_3(t_1, t_2, t_3)$ for some homogeneous polynomials $g_2$ and $g_3$ of degrees $2$ and $3$ respectively (listed in many sources, such as section 9.2 of Dolgachev - Classical algebraic geometry).

In the $A_2$ case, I've tried looking at $(-2)$-curves coming from two consecutive blowups of three collinear points (blow up one point, and then some point on the exceptional divisor in the next blowup). We repeat this for two collinear triples of points. It isn't clear to me how to obtain a sublattice of $R$ (constructed for $X'$) that keeps track of the information above. I think this is what I need to better understand "usual" generators for the root lattice in the smooth case. For example, what do classes in $X'$ coming from classes of lines passing through the singular point in $X$ look like? Are there any suggestions on how to proceed in this situation (or even in the $A_1$ case or other singularity types)?