An obvious ambition of this five-page letter on PT-symmetric quantum
systems is pedagogical. The authors tried to retell the story via
the potentially appealing $N$ by $N$ matrix Hamiltonians $H\neq
H^\dagger$, generalized ``parities" $P$, ``time reversals" $T$ and
``charges" $C$. Personally, I am afraid that they just re-explained
the well known and/or elementary $N < \infty$ results (e.g., of
refs. [7,8]) while leaving their own, ``traditional" and truly
interesting $N=\infty$ (i.e., ODE) examples completely aside. In
technical terms, their two-by-two-matrix expansions (1) in Pauli
matrices prove hardly too illuminating. On phenomenological side,
some of their most interesting formal observations (concerning,
e.g., $C$ in the complex-energy regime) seem still to wait for a
deeper analysis and/or applicability (presumably, {\em outside} of
PT quantum mechanics). Moreover, although the authors are able to
guarantee `` that a positive  $\eta$ exists" via their main
sufficient condition ``that $[C,PT ] = 0$ for {\em all} $C$ that
obey $[C, H] = 0$, $C^2 = 1$", they should have made a comment on
the practical feasibility of such a recipe at large $N \gg 2$. Last
but not least, some of the formulations would deserve more care.
E.g., it is not true that ``if $\eta$ is not positive, then some
eigenvalues are not real and must occur in complex-conjugate pairs".
Indeed, in the light of ref. [7], even if any particular $\eta$
ceases to be positive definite, there may exist, due to its
ambiguity, {\em another}, different, safely positive matrix
$\eta_+$, {\em excluding} the existence of the complex-conjugate
energies.

MR2601810 Bender, Carl M.; Mannheim, Philip D. ${\scr{PT}}$ symmetry
and necessary and sufficient conditions for the reality of energy
eigenvalues. Phys. Lett. A 374 (2010), no. 15-16, 1616--1620. 81Q12