= Solution
Use the <coroot> normalization $\alpha^\vee=2\alpha/(\alpha,\alpha)$, with the pairing understood via the <root system> and its invariant <inner product>. A <dominant integral weight>, called a <dominant weight> here, is a functional $\lambda\in\mathfrak t^*$ such that
$$
\boxed{\langle\lambda,\alpha_i^\vee\rangle\in\mathbb Z_{\geq0}\quad\text{for every simple root }\alpha_i.}
$$
Equivalently, $\lambda=\sum_i n_i\omega_i$ with $n_i\in\mathbb Z_{\geq0}$, where the <fundamental weights> satisfy $\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}$. If dominance and integrality are defined separately, dominance means the inequalities and membership in the <weight lattice> means integrality.
A <highest-weight vector> is a nonzero <weight vector> $v\in V_\lambda$, so $h\cdot v=\lambda(h)v$ for all $h\in\mathfrak t$, annihilated by every positive <root space>:
$$
\boxed{v\ne0,\quad h\cdot v=\lambda(h)v,\quad\mathfrak g_\beta\cdot v=0\quad(\beta\in\Phi^+).}
$$
The last condition is equivalently annihilation by the <raising operators> for the <simple roots>.
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