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] = Σμ (A Bμ/∂xμ - BAμ/∂xμ)
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+aA(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+aA(P)+bB(P+aA(P)), not simply at P+aA(P)+bB(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.

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