All the cube is solved, except for two opposite corners in the LL.

These two corners are permuted properly, but they are disoriented (they are in the right position in relation to the cube but have been rotated).

The corner at BUR is rotated clockwise.
The corner at FUL is rotated counterclockwise.

This algorithm will take care of the orientation of those two corners: (R'F'L') (FR) (F'LF) but it will mess up the edges and permute the other corners.

I need an algorithm that will rotate the corner at BUR counterclockwise and the corner at FUL clockwise, without affecting the rest of the cube.

You can use the commutator (R' D' R D) x2 to rotate corners. Other than that, a specific algorithm to solve this position is F' R D2 R' F U2 F' R D2 R' F U2, though I can't see why you need it.

(F' R D2 R' F U2) x 2 does it. Thanks.

Oh and the reason I need it is in case I ever end up with that configuration after doing F2L. I have actually ended up with that configuration several times.

Hi

A single commutator that does this directly is like so :

[F' D F L D L',U2]=F' D F L D L' U2 L D' L' F' D' F U2

Not the shortest or fastest, but straight forward from "first principles"

-Per

is a commutator when you do something like:

A B C D C' B' A'

sort of like unwinding everything you did up to D?

No. Commutator = A B A' B', where A and B can be either single moves or move sequences. A = (B R' D2 R B') and B = U2 gives (B R' D2 R B') U2 (B R' D2 R B') U2. This is an exception, though, because A = A' but you get the idea.

What you described is a conjugate, C D C'.

See http://www.geocities.com/jaapsch/puzzles/theory.htm

Better to say that A B C D C' B' A' = F C F', where F = A B C.
Avoids notational confusion

-Per

I was actually going to do it that way, but then left it to him to figure out that it's the same thing.

R' U2 R U R' U R L U2 L' U' L U' L'

That twists wrong corners. He asked for opposite corners on same layer

-Per

haha, i learned that one from the rubiks website

That's an interesting alg. Too bad it's not as symmetric as this one: (F' R D2 R' F U2) x 2

i gave him an alg, he has to learn how to mirror it. i didnt want to give him the whole solution. i could have given him a 2 gen that does the same thing.

R' U2 R U R' U R U2 R U2 R' U' R U' R' U2

L U2 L' U' L U' L' U2 L' U2 L U L' U L U2

I don't think I am gonna learn anything just by memorizing algorithms. How do I learn to figure these things out?

how about this one

U' R U R' U' R U L U' R' U R U' R' L'

The first algorithm Adam posted is a combination of two Sunes.

I believe the algorithm from Paiev is a commutator in which the X and the Y move (in XYX'Y') is its own inverse, which allows it to be the exact same sequence twice.

JIMBOB its one that i learned a long time ago (almost 4 years ago) from the rubiks website. i still use it today for BLD and the occasional luckycase last layer.