The Karcher mean of linearly independent triples

We present a classification of 2×2 positive definite Hermitian triples of determinant one having the Karcher mean formula Λ(A,B,C)=A+B+Cdet⁡(A+B+C). We further show that it is equivalent to the trace condition: tr(AB−1)=tr(AC−1)=tr(BC−1) for linearly independent triples.