= Solution
For the unheaded extended-algebra request, fix the central-charge normalization explicitly:
$$
\{Q_\alpha^A,(Q_\beta^B)^\dagger\}=2\delta^{AB}\sigma^\mu_{\alpha\dot\beta}P_\mu,
\qquad
\{Q_\alpha^A,Q_\beta^B\}=2\epsilon_{\alpha\beta}Z^{AB},
\qquad Z^{AB}=-Z^{BA}.
$$
The central charges are scalar numbers on an irreducible massive representation. A <unitary skew-diagonalization of an antisymmetric matrix> puts $Z$ into two-by-two blocks $\begin{pmatrix}0&z_r\\-z_r&0\end{pmatrix}$ and, for odd $\mathcal N$, a leftover zero entry. Since this change of <supercharge> basis is unitary, it preserves the mixed <anticommutator>. Go to the rest frame $P^\mu=(M,0,0,0)$.
For one block set $q_\alpha=Q_\alpha^1$ and $r_\alpha=\epsilon_{\alpha\gamma}(Q_\gamma^2)^\dagger$, with $\epsilon_{12}=1$. Direct use of the algebra gives
$$
\{q_\alpha,q_\beta^\dagger\}=\{r_\alpha,r_\beta^\dagger\}=2M\delta_{\alpha\beta},\qquad
\{q_\alpha,r_\beta^\dagger\}=2z\delta_{\alpha\beta}.
$$
Write $z=|z|e^{i\varphi}$ and form the combinations suggested by the hint:
$$
A_\alpha=\frac{q_\alpha+e^{i\varphi}r_\alpha}{\sqrt2},\qquad
B_\alpha=\frac{q_\alpha-e^{i\varphi}r_\alpha}{\sqrt2}.
$$
Then
$$
\{A_\alpha,A_\beta^\dagger\}=2(M+|z|)\delta_{\alpha\beta},\qquad
\{B_\alpha,B_\beta^\dagger\}=2(M-|z|)\delta_{\alpha\beta},\qquad
\{A_\alpha,B_\beta^\dagger\}=0.
$$
Each diagonal <anticommutator> has nonnegative expectation because it is the sum of squared <norms> of an operator and its adjoint. Therefore $M-|z_r|\ge0$ for every block, proving
$$
\boxed{M\ge\max_r|z_r|.}
$$
If the algebra is instead written with $\{Q,Q\}=\epsilon Z$ without the factor two, the same proof gives $M\ge\tfrac12\max_r|Z_r|$. This is the normalization used by some definitions of the <BPS bound in supersymmetry>; the physical bound is unchanged by the naming of the central charge.
A massive <BPS state> saturates at least one block bound. At saturation $\{B_\alpha,B_\alpha^\dagger\}=0$, so both $B_\alpha$ and $B_\alpha^\dagger$ annihilate every state in its multiplet. The two complex inactive oscillators are four preserved real supersymmetries per saturated block. If $r$ blocks saturate, the preserved fraction is $r/\mathcal N$ and only $2\mathcal N-2r$ complex fermionic oscillators remain active. For a spin-$j$ <Clifford vacuum>,
$$
\boxed{\dim\mathcal H_{\mathrm{BPS}}=(2j+1)2^{\,2\mathcal N-2r},}
$$
compared with $(2j+1)2^{2\mathcal N}$ for a long massive multiplet. Thus <BPS states> have an exact mass-charge relation, preserve part of the <supersymmetry>, and belong to shortened representations. A physical CPT-invariant spectrum may additionally require the conjugate-charge multiplet. The short representation protects the saturated mass relation while it exists; it does not prohibit a decay at a threshold where central-charge phases align and the sum of daughter masses equals the bound.
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