Is this true?

Real matrix diagonalisable over $C \Rightarrow$ diagonalisable over $R$

No — it is false.

The counterexample

This one comes with the Notes pack.

85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Converse assumed

The theorem runs one way. You used it in the other.

See real exam options built on this mistake Drill statements like this

Others that fail the same way

Explained in full in Diagonalisability — the criteria card

From Eigenvalues and Canonical Forms › Diagonalisability criteria

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