Peter Renzland's Easy English Elegant Rubik's Cube Solution:

The Face Positions are: Up, Down, Back, Front, Left, Right.
"R" means "turn the Right face clockwise a quarter turn", etc.

1. Do the Up (white) Corners intuitively, properly aligned.
2. Place the Down (yellow) Corners: Turn the yellow face so that at most 2 corners are
displaced. Hold them between the left thumb and index finger, and swap them:
RBRB RBRB RBRB RBR. (If they are diagonal, make them adjacent for the swap.)

3. Orient the Down (yellow) Corners: If 1 yellow corner cubie is properly aligned,
turn the cube so that it is Down Right Front. Otherwise turn the cube to place a cubie there,
with its yellow facelet on the Right. (The White corners are always Up.)
Do: RBLF RBLF RBLF. (Repeat all of step 3 until the corners are done.)
4. Do the Edges: To move an edge cubie, align it opposite its home cubicle,
properly oriented (down the Back); line up a helper edge-pair (already matched, or both still
random) alongside (down the Right); Do: RRBB RRBB RRBB; invert the line-up turns
(having noted their face-colours).

Intuition assistance for Step 1:

To move a White corner cubie from the Down face into its own cubicle:

Hold the cube so that the target cubicle is at Up-Right-Front.
Turn the Down face to move the cubie to Down-Right-Front.
    If the cubie's White facelet is Down, do   DDD RRR DD R B R BBB.
    If the cubie's White facelet is Right, do   LLL FFF L.
    If the cubie's White facelet is Front, do   B R BBB.
If the cubie is on the UP face, move it to the Down face first.

Intuition assistance for Step 4:

Q: What if 2 edges are flipped in place?
A: Swap them; then send them home properly oriented.

In general, to move a Cubie into a Cubicle, we need to realize that an Edge-Cubicle lies on
2 Faces, and therefore has 2 opposite Cubicles. We need to move the Cubie into one of the
2 opposite Cubicles, properly oriented, or else it will be flipped when we send it into its
target Cubicle. To move a Cubie into a specific Cubicle takes at most 2 turns. But to
move it there properly oriented may take 3 turns. To align a helper edge-pair may take
another turn. Then, holding the cube correctly, we apply the edge-pair swapping Transform
(RRBBRRBBRRBB), and then we undo the line-up turns, in reverse. When remembering the
line-up turns, it helps to refer to them by Colour, not by position.

Keeping the centres correctly aligned may help align the corners and edges.

See also Peter Renzland's Elegant Rubik's Cube Solution Method.

[ peter [at] dancing [dot] org ]