= Solution
Choose the <extended supersymmetry> normalization
$$
\{Q^A_\alpha,\bar Q_{\dot\beta B}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu\delta^A{}_B,\qquad
\{Q^A_\alpha,Q^B_\beta\}=\epsilon_{\alpha\beta}Z^{AB},\qquad \epsilon_{12}=1.
$$
The <central charge in supersymmetry> is antisymmetric in $A,B$, because the full <anticommutator> is symmetric and $\epsilon_{\alpha\beta}$ is antisymmetric. In a massive <one-particle state> rest frame, $P_\mu=(M,0,0,0)$, and the first <anticommutator> becomes $2M\delta_{\alpha\beta}\delta^A{}_B$.
A <unitary skew-diagonalization of an antisymmetric matrix> puts $Z$ into blocks $\begin{pmatrix}0&z_r\\-z_r&0\end{pmatrix}$, with a remaining zero direction if $\mathcal N$ is odd. For one block write $z=|z|e^{i\varphi}$, and label its two supersymmetries $1,2$. The combinations
$$
A_\pm=\frac{Q^1_1\pm e^{i\varphi}(Q^2_2)^\dagger}{\sqrt2},\qquad
B_\pm=\frac{Q^1_2\mp e^{i\varphi}(Q^2_1)^\dagger}{\sqrt2}
$$
have
$$
\{A_\pm,A_\pm^\dagger\}=\{B_\pm,B_\pm^\dagger\}=2M\pm|z|.
$$
For example the mixed contribution to the first expression is $\pm(e^{-i\varphi}z+e^{i\varphi}\bar z)/2=\pm|z|$. This is the phase-aligned version of taking a <supercharge> minus a unitary combination of its adjoints, as suggested by the hint.
For every operator $X$ and normalized state $|v\rangle$,
$$
\langle v|\{X,X^\dagger\}|v\rangle=\|Xv\|^2+\|X^\dagger v\|^2\ge0.
$$
Applying this to $A_-,B_-$ in every block proves \b[the BPS bound]
$$
\boxed{M\ge\frac12\max_r|z_r|.}
$$
If the algebra is instead written with $2Z$ on its right side, the same result is $M\ge\max_r|Z_r|$; the factor is purely a central-charge convention.
A <BPS state> saturates at least one of these inequalities. The corresponding zero-norm combinations of <supercharges> and their adjoints annihilate the representation, so part of <supersymmetry> is preserved. Its <massive supermultiplet> is shortened: each saturated block removes two complex fermionic oscillators, giving $(2j+1)2^{2\mathcal N-2k}$ states from a spin-$j$ <Clifford vacuum> when $k$ blocks saturate, before any necessary <CPT completion of a supermultiplet>. It preserves $4k$ of the $4\mathcal N$ real <supercharges>. Its mass is tied to the <central charges in supersymmetry> and remains protected while it stays on the same short branch. Stability is not unconditional: a <wall of marginal stability> can permit a charged <BPS state> to decay into other <BPS states> with the same total charge.
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