I would start by taking out each different pair of frames to see how effective a 6 frame animation each combination makes. You can use what knowledge you have of key frames to know to leave those. For example, your 1-5 pair is the most important frame of this walkcycle. Don't remove, unless it is just for curioisity. Basically key frames are the frames that contain the full extremes of the movement of a walkcycle, and therefore contain the most information of the movement.
My explanation of keyframes is not complete and possibly not entirely accurate, if someone who is more knowledgable could step in and explain it better.
From there I would pick one combination and edit. Though, you may even find it easier to simply make a 6 framer from scratch. In truth 8 frame animations are not extravagant, but if you really want to take it down that's how I'd start.