[X,Y] = XY – YX
For the Jacobi Identity to be equal to zero, we need the following:
[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0
To prove this, we need to calculate all the terms from the above and then show that they all cancel out. We get a cancellation every time we match +xyz to a -xyz.
Each term on the left of the above equation generates four terms so we will have twelve terms total.
- [X,[Y,Z]]
- X[Y,Z] – [Y,Z]X
- X(YZ-ZY) – (YZ-ZY)X
- XYZ – XZY – YZX + ZYX
- [Y,[Z,X]]
- Y[Z,X] – [Z,X]Y
- Y(ZX-XZ) – (ZX-XZ)Y
- YZX – YXZ – ZXY + XZY
- [Z,[X,Y]]
- Z[X,Y] – [X,Y]Z
- Z(XY-YX) – (XY-YX)Z
- ZXY – ZYX – XYZ + YXZ
XYZ(1) – XZY(2) – YZX(3) + ZYX(4)
YZX(3) – YXZ(5) – ZXY(6) + XZY(2)
ZXY(6) – ZYX(4) – XYZ(1) + YXZ(5)
Rob Sterling 