Solution
= Solution
Each eigenspace of $\alpha_1$ is stable under every commuting $\alpha_i$. Restrict the remaining diagonalizable maps to each eigenspace and induct; concatenating their simultaneous eigenbases proves the result.
= Solution
Each eigenspace of $\alpha_1$ is stable under every commuting $\alpha_i$. Restrict the remaining diagonalizable maps to each eigenspace and induct; concatenating their simultaneous eigenbases proves the result.