Vtyjkyuk

So I'm reading Georgi's Lie Algebras in Particle Physics. With $S$ be a similarity transformation: $D(g) rightarrow D'(g) = S^-1D(g)S$, on page 6, he said:

We will show later that any representation of a finite group is completely reducible.
For example, for $(1.5)$, take
beginequation
tag1.15
S
= frac13
beginpmatrix
1
& 1
& 1
\
1
& omega^2
& omega
\
1
& omega
& omega^2
endpmatrix
endequation
where
beginequation
tag1.16
omega = e^2 pi i /3
endequation
then
beginequation
tag1.17
D'(e)
= left(
beginsmallmatrix
1
& 0
& 0
\
0
& 1
& 0
\
0
& 0
& 1
endsmallmatrix
right)
quad
D'(a)
= left(
beginsmallmatrix
1
& 0
& 0
\
0
& omega
& 0
\
0
& 0
& omega^2
endsmallmatrix
right)
quad
D'(b)
= left(
beginsmallmatrix
1
& 0
& 0
\
0
& omega2
& 0
\
0
& 0
& omega
endsmallmatrix
right).
endequation

(Original scanned image here.)

Here's $(1.5)$:

Here’s another representation of $Z_3$:
beginequation
tag1.5
D(e)
= left(
beginsmallmatrix
1
& 0
& 0
\
0
& 1
& 0
\
0
& 0
& 1
endsmallmatrix
right)
quad
D(a)
= left(
beginsmallmatrix
0
& 0
& 1
\
1
& 0
& 0
\
0
& 1
& 0
endsmallmatrix
right)
quad
D'(b)
= left(
beginsmallmatrix
0
& 1
& 0
\
0
& 0
& 1
\
1
& 0
& 0
endsmallmatrix
right).
endequation

(Original scanned image here.)

I don't see how this $(1.17)$ can be completely reducible. Isn't it equivalent to the $Z_3$ representation in $(1.5)$ above, thus irreducible instead?

Thank you!

Edit: Answer: So since it's not clear $D(g)$ is irreducible or not, we use the similarity transformation $S$ to create an equivalent representation $D'(g)$. We see that $D'(g)$ is in block-diagonal form, and thus is obviously a completely reducible representation. Thus both $D(g)$ and $D'(g)$ are completely reducible.

The representation (1.5) is not irreducible. The vector $e_1 + e_2 + e_3$ is an invariant vector with respect to the representation (1.5), so (1.5) is reducible.

The representation (1.17) is completely reducible, because all of the representing matrices for all the group elements are diagonal. Thus $V = langle e_1 rangle oplus langle e_2 rangle oplus langle e_3 rangle$, and each element of this decomposition is invariant with respect to all group elements.