Exercise 8.C.3

Proof. Define $T\in L({ℂ}^4)$ by

Since $E(7,T)\subset G(7,T),E(8,T)\subset G(8,T)$ and ${ℂ}^4=E(7,T)\oplus E(8,T)$, it follows that $E(7,T)=G(7,T)$ and $E(8,T)=G(8,T)$. We have $\dim <!--\; FUNCTION\; APPLICATION\; -->E(7,T)=\dim <!--\; FUNCTION\; APPLICATION\; -->E(8,T)=2$. Hence, the characteristic polynomial of $T$ is

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