= Solution
Let $\kappa=R/12$, which is constant by the preceding argument. Substitution into <geodesic deviation> gives
$$
(D^2S)^a=\kappa\bigl(T^a g(T,S)-S^a g(T,T)\bigr)=\kappa S^a,
$$
using $g(T,T)=-1$ and the established orthogonality. With $V=DS$, <metric compatibility> yields $\dot f=2g(S,V)$ and
$$
\frac{d}{d\tau}\bigl(g(V,V)-\kappa g(S,S)\bigr)=2g(V,DV)-2\kappa g(S,V)=0.
$$
Therefore
$$
\boxed{K=g(DS,DS)-\frac R{12}g(S,S)\text{ is constant}.}
$$
Both $S$ and $DS$ are orthogonal to $T$, so their squared norms are nonnegative. The minus sign in this <conserved deviation energy in constant curvature> reflects $D^2S=+\kappa S$, rather than an ordinary oscillator when $R>0$.
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