bound on the distance of an isometry to the identity

Question

Consider the space $$ \mathbb{S}^{3,1} := \{ x \in \mathbb{R}^5 : (x_1)^2 + (x_2)^2 + (x_3)^2 +(x_4)^2=1+(x_5)^2 \} $$ with the lorentz metric on $\mathbb{R}^5$ : $$ \langle x,y\rangle _{lor} := x_1 y_1 + x_2y_2 + x_3 y_3 +x_4 y_4 - x_5 y_5 $$ We denote $SO(4,1)$ the group of matrices that preserve this lorentzian scalar product. In particular, they are isometries of $\mathbb{S}^{3,1}$.
On $\mathbb{S}^{3,1}$, I consider the quantity (which is a kind of lorentzian distance) $$ d(x,y) = \inf \left\{ \int_0^1 |\langle \dot{\gamma}(t), \dot{\gamma}(t) \rangle_{lor}|^{1/2} dt\ \Big| \ \gamma \in C^1([0,1]; \mathbb{S}^{3,1}),\ \gamma(0)=x,\ \gamma(1)=y \right\} $$ The question is : is the quantity $\sup_{x\in \mathbb{S}^{3,1}} d(Mx,x)$ finite ? And can we show that there exists $C>1$ such that for any $M\in SO(4,1)$ : $$ C^{-1} |M-I_5| <\sup_{x\in \mathbb{S}^{3,1}} d(Mx,x) < C|M-I_5| $$ where $|\cdot|$ is some matrix norm, and $I_5$ is the identity matrix ?
An other question is : if we have a given Riemannian manifold $(X,g)$ with an isometry $F : X\to X$, can we estimate $\text{dist}(F(x),x)$ in terms of $F$ and any geometric quantity of $X$, in the case where $X$ is a homogeneous space or a Lie group ?