Background:
The standard treatments of the root-systems of $E_6$ and $E_8$ always speak of the roots of $E_6$ as being a subset of the roots of $E_8$, thereby implying that if we construct $E_8$ first, we will get $E_6$ "free-of-charge".
Question:
But what if we start with $E_6$?
We clearly won't get $E_8$ "free-of-charge", because we're missing 240-72 = 168 roots of $E_8$ (the ones which are not in correspondence with roots of $E_6$.)
However, this obvious fact leaves open the question as to whether there is what used to be called an "effective algorithm" for getting $E_8$ from $E_6$.
Is there?
If so, would you take a moment to specify what this algorithm is?
And if not, would you take a moment to explain why not?
Thanks as always for your consideration of this question.
And if you have time, please take a moment to consider the relationship between this question and this one:
$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes