Not every abstract dominant weight integrates

The fundamental weight $\omega$ is dominant in $P$, and it is not an element of $Q=2\mathbb{Z}\omega$; hence for the adjoint form $SO(3)$, whose character lattice is $Q$ by [L2], the weight $\omega$ lies outside $X^{\ast }(T)$.

If some irreducible finite-dimensional representation of $SO(3)$ had highest weight $\omega$, then $\omega$ would be a dominant element of the actual character lattice $X^{\ast }(T)$ by the classification in [L1]; this contradicts step 1.1.

Therefore $\omega$ is a dominant weight of the abstract weight lattice that does not integrate to the compact group form $SO(3)$; the correct statement is the classification by dominant elements of the actual character lattice $X^{\ast }(T)$, between $Q$ and $P$, and the failure is exactly the finite central quotient obstructing the descent of the $SU(2)$-representation of highest weight $\omega$.