The example I'm working off of is $A_2$, so in this case I've determined that the dimension of the Lie algebra is 8, with the dimension of the Cartan subalgebra H being 2. Denoting the basis vectors of the subalgebra as $h_1$ and $h_2$, and the corresponding roots of H* as $\lambda_i$ where i goes from 1 to 6. A defining feature of the Cartan subalgebra is that
$$[h_i, e_j] = ad(h_i)e_j = \lambda_j(h_i)e_j$$
where $e_j$ denotes the other Lie algebra vectors. Therefore the structure constants should be
$$C^i_{ji} = \lambda_j(h_i)$$ (no summation)
But I have no clue what these numbers actually are or how to calculate them.
The lecture series I have been following, https://youtu.be/G9uVcit_VwY?t=4986 seemingly assigns the structure constants the coordinates of a basis vector in R^2, but I'm not sure why. In principle the two fundamental roots have to be 120 degrees apart, so any arbitrary two roots could work as long as it fulfills that condition, meaning you could get an infinite number of different structure constants for $C^i_{ji}$.
In addition, I'm not exactly sure how $h_1$ and $h_2$ are chosen. If $\pi_1$ and $\pi_2$ are the fundamental roots, ie the basis of H*, then the standard choice would be to choose the h's such that
$$\pi_j(h_i) = \delta_{ij}$$
but as far as I understand, this choice is mostly for convenience and not strictly necessary, given that this is a subset of the problem for finding the lambda's, what is the exact process for choosing this basis? Is there a certain degree of freedom we have when choosing?