TITLE: Simple Subalgebras of Simple Lie Algebras
QUESTION [8 upvotes]: Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra of a prescribed type with rank "close" to $n$? For example, if $\mathfrak{g}$ is of type $B_n$ with $n\ge10$, does $\mathfrak{g}$ contain a subalgebra of type $C_{n-2}$?

REPLY [4 votes]: Simple algebras of rank >8 are classical, so you are asking is there a representation (linear, orthogonal or symplectic) of a given dimension of a prescribed Lie algebra. This amounts to the question what is the minimal dimension of a nontrivial linear, orthogonal or symplectic representation of the Lie algebra. A table in Bourbaki answers this.