= Solution
Use the unfolded propagation convention after reflection. Free-space propagation and a <concave spherical mirror> have <matrices>
$$
T(d)=\begin{pmatrix}1&d\\0&1\end{pmatrix},\qquad
F(R)=\begin{pmatrix}1&0\\-2/R&1\end{pmatrix}.
$$
The <mirror> is equivalent in this <paraxial approximation>[paraxial] convention to a <thin lens> of <focal length> $R/2$. The excursion from the reference plane to <mirror> 1 and back is
$$
\boxed{S_1=T(L_1)F(R_1)T(L_1)
=\begin{pmatrix}
1-2L_1/R_1&2L_1-2L_1^2/R_1\\
-2/R_1&1-2L_1/R_1
\end{pmatrix}.}
$$
Likewise $S_2=T(L-L_1)F(R_2)T(L-L_1)$. Multiplying $S_2S_1$, or using cyclic invariance of the <matrix trace>[trace] to move the reference plane, gives
$$
\operatorname{tr}M=2-\frac{4L}{R_1}-\frac{4L}{R_2}
+\frac{4L^2}{R_1R_2}
=4g_1g_2-2,
\qquad g_j=1-\frac L{R_j}.
$$
Consequently the strict criterion of <2/a> is
$$
\boxed{0<g_1g_2<1,\qquad g_1=1-L/R_1,\quad g_2=1-L/R_2.}
$$
The <matrix trace>[trace] is independent of $L_1$, as a change of reference plane only conjugates the <ray transfer matrix>[round-trip matrix]. For the concrete confocal counterexample $R_1=R_2=L$, $L_1=L/2$, each excursion is $\begin{pmatrix}0&L/2\\-2/L&0\end{pmatrix}$ and its square is $-I$. This gives $g_1g_2=0$ and the degenerate boundary exception already explained.
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