Simple Subalgebras of Simple Lie Algebras 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 $C_{n-2}$?
 A: 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.
A: I am not sure if this is exactly what you are looking for, but there have been some classic works, developing general methods for such topics: 


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*In Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. (N.S.), 1952,    Volume 30(72),  Number 2, p. 349–462, the author -among others- classifies (up to linear equivalence) the simple Lie subalgebras of the Lie algebras of exceptional type. The original paper is in Russian (and is free) but you can find a translation in the translation series of AMS here (not free). Some of these results have been refined and generalized in a 2006 paper by Minchenko (but i do not have available the exact reference right now), 

*In M. Lorente, B. Gruber, Classification of Semisimple Subalgebras of Simple Lie Algebras, Journal of Mathematical Physics 13, 1639 (1972), the authors extend Dynkin's methods and classify the simple subalgebras of Lie algebras of classical type of rank $\leq 6$, up to linear equivalence, 

*In W. A.de Graaf, Constructing semisimple subalgebras of semisimple Lie algebras, Journal of Algebra, v. 325, 1, 2011, p.416-430, general methods are obtained for classifying the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. Some of these methods are then applied to obtain a classification of the semisimple subalgebras of the simple Lie algebras of rank $\leq 8$. 

A: The answer to the question in your example is no in general: $B_n$ does not contain $C_{n-2}$ for large $n$. To see this, observe that $B_n$ has an irreducible orthogonal representation $V$ of dimension $2n+1$. The Weyl dimension formula shows that for any simple Lie algebra, the dimension of an irreducible representation is the smallest for fundamental representations. 
[Edited] 
For large $n$, the smallest dimensional fundamental representation of $C_{n-2}$ is the standard one, of dimension $2n-4$. 
Therefore  the restriction of $V$ to $C_{n-2}$ can only be some copies of the trivial representation and ONE copy of the standard representation $W$ of dimension $2n-4$. This means that the group of type $C_{n-2}$ must preserve both a symplectic form and a quadratic form on $W$, which is impossible by Schur's lemma.  Hence  the restriction of $V$ to $C_{n-2}$ can only be a sum of copies of the trivial representation which is 
impossible.
I had previously given a wrong reason (assuming that the rep of $B_n$ had dimension $n$; thanks to @BS for pointing this out. 
