\cQ^a, \mathcal{O}^b(x) ] = i \varepsilon^{abc} \mathcal{O}^c(x)
.$$ Defining the usual isospin raising and lowering operators $T^\pm = T^1 \pm i T^2$, we have $$\label{eq:key}
\mathcal{O}^\pm (x) = \mp [ \cQ^\pm, \mathcal{O}^3(x)]
.$$ These relations enable us to relate isospin changing matrix elements to isovector ones.

As an example, consider the neutron to proton matrix element of $\mathcal{O}^+$. We have $$\label{eq:protonkey}
\langle p \, | \, \mathcal{O}^+  | \, n \rangle 
= 
\langle p \, | \, \mathcal{O}^3  | \, p \rangle
- 
\langle n \, | \, \mathcal{O}^3  | \, n \rangle 
,$$ by virtue of Eq. . This result can be rewritten in various forms using the isoscalar combination $$\mathcal{O}^I(x) = \ol Q(x) \mathbbm{1} \, \Gamma \, Q(x)
,$$ and the fact that $$\langle p \, | \, \mathcal{O}^I  | \, p \rangle
- 
\langle n \, | \, \mathcal{O}^I  | \, n \rangle
= 
0 
.$$ For example, $\langle p \, | \, \mathcal{O}^+  | \, n \rangle 
= 2 \langle p \, | \, \mathcal{O}^3  | \, p \rangle
= \langle p \, | \, \ol u \, \Gamma \, u  | \, p \rangle - \langle n \, | \, \ol u \, \Gamma \, 