Equivalent definitions of CM abelian varieties

Question

I am reading Milne's notes on CM (page 27) https://www.jmilne.org/math/CourseNotes/CM.pdf

He defined a CM abelian variety $A$ to satisfy $$2\dim A=[\text{End}^{0}(A):\mathbb{Q}]_\text{red}.$$ Then followed a proposition saying

$A$ is CM $\leftrightarrow$ $\text{End}^{0}(A)$ contains an etale subalgebra of degree $2\dim A$.

In the proof, he showed that the degree of every maximal etale subalgebra is $[\text{End}^{0}(A):\mathbb{Q}]_\text{red}$.

But how can we go $\leftarrow$ direction? We only know there exists (not necessarily maximal) a $2\dim A$ etale subalgebra. Am I missing something?

Thanks!

Answer 1

By 1.3 and 3.1 of the notes, we know that

(degree of any semisimple subalgebra of $End^{0}(A)$)$ \leq\lbrack End^{0}(A)\colon\mathbb{Q}]_{red}\leq2 dim(A)$

from which the statement follows.