Can any(some) generator(s) of adjoint representation of SU(3) (Gell-Mann construction) be reducible under SU(2)?

Question

I was going through another thread on MathStack which was dealing with a specific question from Georgi, Lie Algebras in Particle Physics. I am not specifically looking for an answer/clarification with regards to Particle Physics. Also, the thread came off as somewhat abstract and in the accepted answer I happened to chance across terms which were unexplained. I am putting the question here again. I have answered the first chunk of the question apart from the adjoint part. I haven't seen any resource explain reducibility under a sub-representation.

Show that $T_1$ , $T_2$ and $T_3$ generate an SU(2) subalgebra of SU(3). Every representation of SU(3) must also be a representation of the subalgebra. However, the irreducible representations of SU(3) are not necessarily irreducible under the subalgebra. How does the the representation generated by the Gell-Mann matrices transform under this subalgebra. That is, reduce, if necessary, the three dimensional representation into representations which are irreducible under the subalgebra and state which irreducible representations appear in the reduction. Then answer the same question for the adjoint representation of SU(3).

I am unable to comprehend as to how to address the adjoint part in depth. I have been seeing quite a lot of hand waving arguments as to how the reduction may be done, but I guess I am having difficulty understanding how to show it either through brute force adjoint representation, or by using adjoint generator transformations.

Now, say, I have the three adjoint representation generators of SU(3),constructed by hand, which I did, how do I show that they are a Kronecker product of two irreducible representations of lower dimensions, under SU(2)

Answer 1

Probably of no use to OP at this point but for anyone who might stumble in here like I did, you can "brute force" the problem by putting the adjoint $T_1$, $T_2$, and $T_3$ into block diagonal form with a similarity transformation. Take

$S = \frac{1}{\sqrt{2}}\begin{pmatrix}-i & 0 & -i & & & & & \\ 0 & \sqrt{2} & 0& & & & & \\ -1 & 0 & 1 & & & & & \\ & & & 0 & -1 & -1 & 0 & \\ & & & -i & 0 & 0& -i & \\ & & & 0 & i & -i & 0 & \\ & & & 1 & 0 & 0& -1 & \\ & & & & & & & \sqrt{2}\end{pmatrix}$

to see how they break down into $SU(2)$ irreducible representations (it's not a product of two by the way, but a direct sum of four). As for the method to derive something like this on your own, I don't have a great answer. I can list the steps I went through though.

1. Pick one to diagonalize (I chose $T_2$)
2. Take inventory from its eigenvalues to get an idea of what representations you can make from them
3. Make a similarity matrix from the normalized eigenvectors
4. Trial & error with the phase factors of each vector until a representation from your inventory appears in the block diagonal