Show that $A$ is similar to a diagonal matrix iff $b=c=d=e=f=g=0$

Show that $A$ is similar to a diagonal matrix iff $b=c=d=e=f=g=0$

Show that $A$ is similar to a diagonal matrix iff $$b=c=d=e=f=g=0$$ $$A=
\left(\begin{array}{cccc}a & b & c & d \\ 0 & a & e & f\\ 0 & 0 & a & g\\
0 & 0 & 0 & a \end{array}\right)$$
Attempt: If $$b=c=d=e=f=g=0$$, then clearly $A$ is similar to a diagonal
matrix. $$A = I^{-1}AI$$ Conversely, if $A$ is similar to a diagonal
matrix, then $$A=P^{-1}DP$$ where $P$ is non-singular. Suppose
$$b,c,d,e,f,g \neq 0$$ We know that the only eigenvalue of $A$ is $a$. To
be diagonalizable, we know that $A$ must have 4 eigenvectors. With the
assumption of $$b,c,d,e,f,g \neq 0$$ we get that the only eigenvector
w.r.t. the eigenvalue $$\lambda=a$$ is $$(1,0,0,0)$$ This contradicts the
fact that $A$ is diagonalizable. Hence $$b=c=d=e=f=g=0$$
Is the above logic wrong? Is there any other way I can prove this?