Subgraphs of Dynkin Diagrams

Question

Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g} $ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras $\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ iff $A$ is a subgraph of $B$?

Answer 1

No. For example, the Lie algebra of type $D_4$ is the algebra $\mathfrak{so}_8$ of $8\times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $\mathfrak{sl}_8$ of $8\times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.