Orthogonal Clifford families and centriole spaces

Let p be a positive even integer and define SO(p)={J∈M_p(ℝ):JᵀJ=I, det J=1}. Its trace metric is the left-invariant Riemannian metric whose inner product at the identity is ⟨A,B⟩=−(1/p)tr(AB) for skew-symmetric A,B; at J its value on tangent vectors JA,JB is the same. An orthogonal negative-square Clifford family of length l is a list J₁,…,J_l∈SO(p) with J_i²=−I and J_iJ_j=−J_jJ_i for i≠j, equivalently a representation of C_l=Cl(ℝ^l,−‖·‖²) by these matrices. For a Riemannian manifold P and points x,y, its midpoint set is {γ(1/2): γ:[0,1]→P is a length-minimizing geodesic from x to y}; a centriole is a connected component of such a midpoint set. Put P₀=SO(p), J₀=I, and for 1≤j≤l define P̂_j={J∈SO(p):J²=−I and JJ_i=−J_iJ for 1≤i<j}; let P_j be the connected component of P̂_j containing J_j. A compatible family also satisfies J_j∈P_(j−1). The geometric midpoint interpretation of this algebraic component is the claim of the next lemma.