Consider the Lie algebra $so(n)$ equipped with the metric $\langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}$. Similarly equip the tangent space at other points of $SO(n)$ by left translation. My question is, is the exponential map $\exp: so(n) \to SO(n)$ 1-Lipschitz? I can show it's Lipschitz using the following result from Wikipedia:

$$ \| e^{X+Y} - e^Y \| \le e^{\|X\|} e^{\|Y\|} \|X\|$$ for any matrix norm $\|\bullet \|$. But this yields an optimal Lipschitz constant of 2, by rescaling the Hilbert Schmidt norm. I really need the constant to be $1$. Maybe it's not even true?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\so{\mathfrak{so}}$Consider the Lie algebra $\so(n)$ equipped with the metric $\langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}$. Similarly equip the tangent space at other points of $\SO(n)$ by left translation. My question is, is the exponential map $\exp: \so(n) \to \SO(n)$ 1-Lipschitz? I can show it's Lipschitz using the following result from Wikipedia:

$$ \| e^{X+Y} - e^Y \| \le e^{\|X\|} e^{\|Y\|} \|X\|$$ for any matrix norm $\|\bullet \|$. But this yields an optimal Lipschitz constant of 2, by rescaling the Hilbert Schmidt norm. I really need the constant to be $1$. Maybe it's not even true?