  The non-trivial complete totally geodesic submanifolds of the complex
hyperbolic plane $\mathbb H_{\mathbb C}^2$ are the complex geodesics and the
real planes. We present two new proofs for this fact. One is a short proof
based on an algebraic formula for the Riemann curvature tensor due to S.
Anan'in and C. Grossi and resembles the traditional proof using Lie theory. The
other is purely elementary and geometric, relying on the structures in $\mathbb
H_{\mathbb C}^2$ instead of general theories. In this second approach, we prove
a slightly stronger result: the only non-trivial complete totally geodesic
subsets of $\mathbb H_{\mathbb C}^2$ are the complex geodesics and the real
planes without assuming that the subsets are submanifolds a priori. This second
proof is also intriguing for only making use of elementary geometric
constructions.
