= Solution
The linear predictor is unchanged by $\alpha_j\mapsto\alpha_j+c$, $\beta_t\mapsto\beta_t+d$, $\mu\mapsto\mu-c-d$. Without constraints, different parameter triples therefore give the same rates and <likelihood>. <Identifiability> requires choosing one representative of this redundancy.
Taking the first levels as reference yields $\alpha_1=\beta_1=0$. Then
$$
\mu=\log\lambda_{11},\qquad
\alpha_2=\log\lambda_{21}-\log\lambda_{11},\qquad
\beta_2=\log\lambda_{12}-\log\lambda_{11}.
$$
These identify all three free parameters, while $\lambda_{22}=\lambda_{21}\lambda_{12}/\lambda_{11}$ is the model's no-interaction constraint. \b[The zero reference effects fix a parametrization; they do not remove the baseline rate or impose equal rates across groups.]
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