  We argue for Brackets Consistency to be a `Pillar of Geometry', i.e. a
foundational approach, other Pillars being 1) Euclid's constructive approach,
2) the algebraic approach, 3) the projective approach, and 4) the geometrical
automorphism groups `Erlangen' approach. We proceed via a distinct Pillar 5) -
Killing's computational route to Erlangen's groups - firstly by picking out the
ensuing geometrically significant brackets subalgebras as consistent
possibilities. Secondly, by considering which generators, arising piecemeal
from different generalized Killing equations, combine brackets-consistently,
recovering the affine-conformal and projective-conformal alternatives. Thirdly,
we cease to rest on Killing's Pillar altogether, replacing it by mere
polynomial ans\"{a}taze as source of generators. Even in this case - demanding
(Lie) Brackets Consistency - suffices to recover notions of geometry, as
indicated by the below alternative just dropping out of the ensuing brackets
algebra, and is thus declared to be Pillar 6. The current article is further
motivated, firstly, by the gradually accumulating successes in reformulating
Theoretical Physics to run on the Dirac Algorithm for classical constraint
algebraic structure consistency. Secondly, this is sufficiently driven by
Brackets Consistency alone to transcend to whichever context possesses such
brackets. Thirdly, we derive the conformal-or-projective alternative in
top-automorphism-group $d \geq 3$ Flat Geometry. In a manner by now familiar in
Dirac Algorithm applications, a) these two top geometries moreover arise side
by side as roots of a merely algebraic quadratic equation. b) This equation
arises from requiring that the self-bracket of a general ansatz for generators
be strongly zero. c) For, if not, an infinite cascade of generators would
follow, by which the ansatz would not support a finite Shape Theory.
