The Lie bracket, sometimes called the breaker of quadrilaterals, is a mathematical function taking two vector fields and returning a third. Being a ubiquitous if not utterly fundamental operation, it gets brief notation: [A,B]. The resemblance to the commutator is not entirely coincidental, as we shall see.

The definition is

[`A`,`B`] = Σ_{μ} (`A` ∂`B`^{μ}/∂x^{μ} - `B` ∂`A`^{μ}/∂x^{μ})

with the μ being the indices of the dimensions of the vector field and x the basis vectors. These partial derivatives should be the covariant derivatives, not the naive form.

What does this mean? Well, start at some known point, say, `P`. Take the value of A -- remember, it's a vector -- and go where it points (times some differential factor `a`). Now you're at `P`+`a``A`(`P`). Now take the value of B and go where it points (times some differential factor `b`).

Here is where it gets complicated. We are now at `P`+`a``A`(`P`)+`b``B`(`P`+`a``A`(`P`)), not simply at `P`+`a``A`(`P`)+`b``B`(`P`).

To isolate the complexity from the boring parts, we turn this around and go back a different way: go along -aA, then -bB (evaluated locally each time). If A and B are entirely homogeneous, never changing value for the entire traversal, you will end where you started -- you will have drawn a closed quadrilateral.

However, if they DO change, you will end up at `P`+`ab`[`A`,`B`] instead of `P`. The quadrilateral is only unbroken if the Lie bracket is zero, hence the nickname.

If you do this for some macroscopic distance, you will need to include higher-order correction terms.

Other tidbits of note:

The Lie Bracket is antisymmetric under swapping arguments.

The Lie Bracket is coordinate-invariant, which makes it especially useful in situations where that's hard to achieve, such as in curved space (in which the coordinate system itself may have a nonzero Lie bracket).

Compare to the error of parallel transport, in which a crab-walker going in loops similar to this one ends up turning despite eir best efforts.