Example 1.

First initialize the jet space for 2 independent variables and 2 dependent variables and prolong it to order 4.

Define 2 vector fields X1 and Y1.

Compute the generalized Lie bracket of X1 and Y1.

We show how this result is obtained.  First prolong X1 to the order of the coefficient in Y1, namely 2.  Apply the prolonged vector field to the coefficient of Y1.

Next prolong Y1 to the order of the coefficient in X1, namely 4.  Apply the prolonged vector field to the coefficient of Y1.

The difference between term1 and term2 gives the coefficient of the generalized Lie bracket of X1 and Y1.

Example 2.

The generalized Lie bracket is not restricted to evolutionary (vertical) generalized vector fields.

Example 3.

The generalized Lie bracket for a pair of 1st order evolutionary vector fields coincides with the Jacobi bracket.  For example:

DifferentialGeometry, JetCalculus, AssignVectorType, LieBracket, LieDerivative, Prolong