Figured it Out

The bit at the end of the last post where I was worried about the commutator (lie bracket in the lie algebra of vector fields, not vectors at the identity) of two left-invariant vector fields not being itself left-invariant was just a mistake. I shouldn't expect [extend(d/dtheta), extend(d/dphi)] = d/dpsi, but rather [extend(d/dtheta), extend(d/dphi)] = extend(d/dpsi)! If you do that, then it works out:

(test-case 
 "Extended tangent vectors are really left-invariant"
 (check-tuple-close? 
  1e-6
  ((vector-field->component-field 
    (- ((lie-algebra-bracket SO3) d/dphi d/dtheta)
       ((natural-extension SO3) d/dpsi))
    SO3-rectangular-chart)
   ((slot-ref SO3-rectangular-chart 'chiinv)
    (up 0.02345453 0.0349587 0.0435897)))
  (up 0 0 0)))

passes with flying colors. (In case the above code is not completely clear---irony of the century---it's computing the components of the vector field ([extend(d/dtheta),extend(d/dphi)]-extend(d/dpsi)) at some arbitrary point in SO3 and verifying that they're zero.