On the other hand, it is obvious that the six basic face movements can solve the cube. This fact is hardly useful because the face movements themselves affect too many pieces each time. Therefore, they cannot help us to achieve meaningful goals.
Notice that corners always move to corners and edges to edges. You can never move a corner to an edge. Therefore, they are very different objects, and should be treated separately. The minimum basis vectors I have chosen are
CM3 moves 3 corners, preserving other corners
CT2 twists 2 corners, preserving other corners
EM3 moves 3 edges, preserving all other pieces
EF2 flips 2 edges, preserving all other pieces
Since the Strategy of 8 Corners settles corners first, corner algorithms can affect edges, but edge algorithms must preserve corners.
These basis algorithms, together with their mirror images and conjugates (see below), are sufficient to solve the cube. Notice that by looking at the cube as corners and edges, you have to handle only two scenarios how to work with corners, and how to work with edges. For each scenario, you have only two things to do move them around, or fliptwist them in place. That is all you need to solve the cube.