I fail to understand an elementary statement about roots of a semisimple Lie algebra $\mathfrak{g}$ and weights of a faithful representation.
There are two sources for the classification of simple Lie superalgebras: The article "Lie superalgebras" by Kac and the book "The theory of Lie superalgebras - an introduction" by Scheunert. The following assertion occurs in the article by Kac (Proof of Lemma 1.4.1) and the book by Scheunert (Proof of lemma 2 in the appendix):
Let $\rho$ be a finite-dimensional irreducible faithful representation of a semisimple Lie algebra $\mathfrak{g}$. Let $\lambda$ be the highest weight of $\rho$ and $\mu$ the lowest weight. Then $2\lambda$ or $\lambda - \mu$ can only be roots of $\mathfrak{g}$ if $\mathfrak{g}$ is simple.
Kac doesn't give any arguments. Scheunert writes:
"It is well-known that $-\mu$ is the highest weight of the representation contragredient to $\rho$; in particular $-\mu$ is dominant. Recalling that $\rho$ is faithful we conclude that $2\lambda$ or $\lambda - \mu$ can be a root of $\mathfrak{g}$ only if $\mathfrak{g}$ is simple".
I agree that $-\mu$ is dominant, but why does this imply the claim?