  A local resolution of the Problem of Time has recently been given, alongside
reformulation as a local theory of Background Independence. The classical part
of this requires just Lie's Mathematics, much of which is basic: i) Lie
derivatives to encode Relationalism. ii) Lie brackets for Closure giving Lie
algebraic structures. iii) Observables defined by a Lie brackets relation, in
the constrained canonical case as explicit PDEs to be solved using Lie's flow
method, and themselves forming Lie algebras. iv) Lattices of constraint
algebraic substructures induce dual lattices of observables subalgebras.
  The current Article focuses on two pieces of `higher Lie Theory' that are
also required. Preliminarily, we extend Dirac's Algorithm for Constraint
Closure to `Lie's Algorithm' for Generator Closure. 1) We then reinterpret
`passing families of theories through the Dirac Algorithm' - a method used for
Spacetime Construction (from space) and getting more structure from less
structure assumed more generally - as the Dirac Rigidity subcase of Lie
Rigidity. We also provide a Foundations of Geometry example of specifically Lie
rather than Dirac Rigidity, to illustrate merit in extending from Dirac to Lie
Algorithms. We point to such rigidity providing a partial cohomological (and
thus global) selection principle for the Comparative Theory of Background
Independence. 2) We finally pose the universal (theory-independent) analogue of
GR's Refoliation Invariance for the general Lie Theory: Reallocation of
Intermediary-Object Invariance. This is a commuting pentagon criterion: in
evolving from an initial object to a final object, does switching which
intermediary object one proceeds via amount to at most a difference by an
automorphism of the final object? We argue for this to also be a selection
principle in the Comparative Theory of Background Independence.
