1. The problem statement, all variables and given/known data

Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as:

5 →4⊕1
10→6⊕4

2. Relevant equations

3. The attempt at a solution

I understand that 5 is rep of SO(5) corresponding to Dynkin labels (1, 0). 1 is of course trivial representation. But what type of rep is 4? Once I started calculating weights I found out that Dynkin labels corresponding to (2, 0) as well as (1, 1) for SO(4) both have 4 weights (and different ones). And none of them seem to reproduce 5! Maybe there is an easier way, how would you show these decomposition?

Spoiler: Results I've got

Weights for 5 [Dynkin (1,0) for SO(5)]:
(1, 0), (-1, 2), (0,0), (1, -2) (0, -1)

Weights for 4? [Dynkin (1,1) for SO(4)]:
(1,1), (-1, 1), (1, -1), (-1, -1)

Weights for 4? [Dynkin (3, 0) for SO(4)]:
(3, 0), (1, 0), (-1, 0), (-3, 0)

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?