Explicitly constructing subgroups of exceptional algebraic groups

There is now a complete classification of the Lie primitive (i.e., not contained in a positive-dimensional subgroup) subgroups of the exceptional algebraic groups of types G₂, F₄ and E₆. This is true, in the sense that there is a table in a paper. For G₂ there is also a Magma computer program that, given a prime p, returns as 7x7 matrices all Lie primitive subgroups of G₂ in characteristic p, written over a minimal subfield of an algebraically closed field. No such program exists for F₄ (or the maximal subgroups of the large Ree groups ²F₄(q) ) or E₆.

This computational mathematics project would produce constructive programs that work fast (the program for G₂ finds all Lie primitive subgroups in less than a second for 20-digit primes) giving these subgroups as explicit matrices over finite fields. The final aim is to produce a Magma package where the user can call F4maximals(q) and be given a complete list of all maximal subgroups of the finite group F₄(q), a moderately more ambitious goal than just the Lie primitive subgroups. The same would happen for the other series of groups of Lie type.