  For F an algebraic extension of Q2, the conjugacy classes of invertible,
2-by-2, trace-zero matrices under the action of G := SL2(F) are analyzed
relative to the quadratic extension that splits the respective characteristic
polynomial. The stabilizer in G of each such matrix is computed as a stabilizer
of a simple, Galois invariant path in the Bruhat-Tits Tree of G.
