Determining the basis of a Lie algebra may be a long and tedious process. The combinations of Lie brackets in Example 15.17 were given; however, it is not known in advance which ones will produce independent vector fields. Numerous Lie brackets may be needed, including some that are nested, such as . The maximum depth of nested Lie bracket operations is not known a priori. Therefore, a systematic search must be performed (this can in fact be modeled as a discrete planning problem) by starting with , , and iteratively generating new, independent vector fields using Lie brackets.

One popular approach is to generate the *Philip Hall basis* (or
*P. Hall basis*) of the Lie algebra
. The
construction of the basis essentially follows breadth-first search, in
which the search depth is defined to be the number of nested levels of
bracket operations. The *order* (or *depth*) of a Lie
product is defined recursively as follows. For the base case, let
for any of the system vector fields. For any Lie product
, let

(15.108) |

Thus, the order is just the nesting depth (plus one) of the Lie bracket operations. For example, and .

In addition to standard breadth-first search, pruning should be
automatically performed to ensure that the skew symmetry and
Jacobi identities are always utilized to
eliminate redundancy. A *P. Hall basis*
is a sequence,
, , , of Lie products
for which:

- The system vector fields are the first elements of .
- If , then .
- Each if and only if: a) and , and b) either for some or for some such that .

, | , | , | |

, | , | , | |

, | , | , | , |

, | , | , | . |

(15.109) |

Note that all of the Lie products given here may not be linearly independent vector fields. For a particular system, linear independence tests should be performed to delete any linearly dependent vector fields from the basis.

When does the sequence terminate? Recall that
can be no greater than , because
. In other words, at every state , the number of possible independent velocity vectors is no more
than the dimension of the tangent space at . Therefore,
can be terminated once independent vector fields are obtained
because there is no possibility of finding more. For some systems,
there may be a depth after which all Lie brackets are zero. Such
systems are called *nilpotent* of order
. This occurs, for example, if all components of all vector fields
are polynomials. If the system is not nilpotent, then achieving
termination may be difficult. It may be the case that
is strictly less than , but this is usually
not known in advance. It is difficult to determine whether more Lie
brackets are needed to increase the dimension or the limit has already
been reached.

Steven M LaValle 2020-08-14