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For the convention in cross-ratio normalization of four projective points, permuting four distinct projective points produces the six displayed rational expressions. Interchanging the first pair sends $\lambda$ to $1/\lambda$; interchanging the middle pair sends it to $1-\lambda$. The four permutations consisting of the identity and the three double transpositions form a Klein four-group preserving the cross-ratio. The resulting six-element action gives every displayed expression.

For real $\lambda \notin \{0,1\}$, repeated values occur exactly for $\lambda \in \{-1,1/2,2\}$, and then the orbit has size three. To prove this, a repetition gives a nonidentity stabilizer after composing one operation with the other's inverse. Its fixed-point equations give $-1$, $1/2$, $2$, or ${\lambda }^2-\lambda +1=0$, with the last case nonreal. With one point at infinity, $\lambda =(x_1-x_3)/(x_1-x_2)$; the three real exceptional values mean that one finite point is the midpoint of the other two. Thus these harmonic conjugate points configurations have three collinear equally spaced finite points in that affine coordinate.

The needed general-position theorem is sharp triple transitivity of the projective linear group on the projective line: for any two ordered triples of distinct projective points, exactly one projective transformation takes the first triple to the second.

In particular, let $T_P$ take $(P_0,P_1,P_2)$ to $(\infty ,0,1)$. In these coordinates,so cross-ratio invariance gives $T_P(P_3)=\lambda$, where $\lambda =(P_0{P}_1:P_2{P}_3)$. Distinctness implies $\lambda \ne 0,1$ and $\lambda$ is finite. The entire ordered quadruple is thus normalized toFor a second quadruple $(Q_0,Q_1,Q_2,Q_3)$ with the same cross-ratio, its normalizing map $T_Q$ yields the same quadruple. ConsequentlyThis proves the equivalence. Uniqueness on the first three points also shows that the projectivity carrying one ordered quadruple to the other is unique. The cross-ratio normalization of four projective points therefore gives a complete invariant for this ordered classification.