R. Westwick’s convexity theorem for the numerical range [Linear Multilinear Algebra 2, 311-315 (1975; Zbl 0303.47001)] is generalized in the context of compact connected Lie groups. Let \(G\) be a compact Lie group with Lie algebra \(\mathfrak g\) which is equipped with a \(G\)-invariant inner product \(\langle\cdot,\cdot\rangle\). For \(X_1,X_2,Y\in \mathfrak g\), the \(Y\)-numerical range of \((X_1,X_2)\) is defined to be the following subset of \({\mathbb R}^2\): \(W_Y(X_1,X_2)=\{(\langle X_1,\text{Ad} (g),Y\rangle\), \(\langle X_2,\text{Ad} (g),Y\rangle)\): \(g\in G\}\). The convexity of \(W_Y(X_1,X_2)\) is proved via Aityah’s lemma on compact connected symplectic manifolds and Kirillov-Kostant-Souriau symplectic structure of the co-adjoint orbits of a Lie group. Moreover, convexity is proved for classical groups as \(SO(n)\), \(SU(n)\) and \(Sp(n)\), though it fails to be true with \(G=O(2n)\), but remains valid when \(G=O(2n+1)\).