# How to make a triplet out of 2 doublets in the $SU(2)$ representation?

In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $$\phi_a$$ and $$\phi_b$$, under $$SU(2)$$ symmetry

the law is $$2 * 2 = 1 + 3$$ and to produce a triplet under $$SU(2)$$ out of 2 doublets there is another formula: $$(\phi_a\phi_b)_{ij}=(\phi_a)_i(\phi_b)_j-\frac{\delta_{ij}}{2}(\phi_a)_k(\phi_b)_k$$

How do I write it explicity this 3 dimensional vector if for instance $$\phi_a=\binom{A}{B}$$ and $$\phi_b=\binom{C}{D}$$, the indices made me very confused.

In a product of two doublets $$(A\ B)$$ and $$(C\ D)$$, the components of the triplet are proportional to $$AC$$, $$AD+BC$$, and $$BD$$ (omitting normalization factors). The singlet is $$AD-BC$$.