= Solution
The same <invariant subspaces> work for $S=\alpha I+\beta T$: it acts by $\alpha$ on $\ker T$ and by $\alpha+\beta$ on $\operatorname{im}T$. Its <minimal polynomial> is therefore the product of the distinct <linear polynomials> belonging to the nonzero summands. Explicitly,
$$
\boxed{m_S(x)=\begin{cases}
x-\alpha,&\beta=0\text{ or }T=0,\\
x-(\alpha+\beta),&\beta\ne0,\ T=I,\\
(x-\alpha)(x-\alpha-\beta),&\beta\ne0,\ T\ne0,I.
\end{cases}}
$$
When $\beta\ne0$, the two candidate <eigenvalues> are distinct in any <field>. This is the <minimal polynomial of an affine function of a projection>.
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