We know that the Lie algebra $$usp_4 \sim sp_2 \sim so_5.$$
It is said in https://drive.google.com/file/d/1kFuSDO_3apXes45FH3YPa4s4HEZd9Vj9/view:
"There are two types of subalgebras of simple Lie algebras defined by E.B. Dynkin. One is an R-subalgebra; the other is an S-subalgebra. An R-subalgebra is a subalgebra that contains several regular subalgebras, where all R-subalgebras is obtained by deleting dots from (extended) Dynkin diagrams. An S-subalgebra is a subalgebra that is not any R-subalgebra. It can be found by using dimensions and type of irreducible representations of a Lie algebra and its subalgebra. (Some S-subalgebras can be found by using (extended) Dynkin diagrams but it is difficult to find almost all S-subalgebras. The rank of an S-subalgebra is always smaller than the rank of its original Lie algebra.)"
question: I am not sure why for example by the branch rule from $usp_4 \sim sp_2 \sim so_5$ to $su_2$ gives several different results?
See Table F.3.2.1
usp_4 to su_2 (S): $$ 4 = 4, $$ $$ 5=5 $$
But See Table F.3.1.1
usp_4 to su_2 + su_2 (R): $$ 4 = (2,1)+(1,2), $$ $$ 5= (2,2)+(1,1). $$
But See Table F.3.1.2 usp_4 to su_2 + u_1 (R): $$ 4 = 2_1+2_{-1}, $$ $$ 5= 3_0 + 1_2 + 1_{-2}. $$
Why are they different? and how R and S Lie subalgebra related?
