## 2 Notation and Background
### Lie Groups  
Let \(G\) denote a connected Lie group of dimension \(d\), i.e., a smooth manifold with a group structure such that the group operations \(G\times G\
i(x,y)\stackrel{{ H}}{{\mapsto}}xy\in G\) and \(G\
i x\stackrel{{ L}}{{\mapsto}}x^{-1}\in G\) are smooth maps. If \(x\in G\), the left-multiplication map, \(L_{xy}\), defined by \(y\mapsto\mu(x,y)\), is a diffeomorphism from \(G\) to itself. Similarly, the right-multiplication map \(R_{x}y\) defines a diffeomorphism from \(G\) to itself by \(y\mapsto\mu(y,x)\). Let \(dL_{x}\colon TG\to TG\) denote the pushforward map given by \((dL_{x})_{y}\colon T_{y}G\to T_{xy}G\). A vector field \(V\) on \(G\) is said to be left-invariant if \((dL_{x})_{y}V(y)=V(xy)\). The space of left-invariant vector fields is linearly isomorphic to \(T_{e}G\), the tangent space at the identity element \(e\in G\). By equipping the tangent space \(T_{e}G\) with the Lie bracket, we can identify the Lie algebra \(\mathfrak{g}\) with \(T_{e}G\). The group structure of \(G\) makes it possible to define an action of \(G\) on its Lie algebra \(\mathfrak{g}\). The conjugation map \(C_{x}:=L_{x}\circ R_{x}^{-1}\colon y\mapsto xyx^{-1}\), for \(x\in G\), fixes the identity \(e\). Its pushforward map at \(e\), \((dC_{x})_{e}\), is then a linear automorphism of \(\mathfrak{g}\). Define \(\operatorname{Ad}(x):=(dC_{x})_{e}\), then \(\operatorname{Ad}:x\mapsto\operatorname{Ad}(x)\) is the adjoint representation of \(G\) in \(\mathfrak{g}\). The map \(G\times\mathfrak{g}\
i(x,v)\mapsto\operatorname{Ad}(x)v\in\mathfrak{g}\) is the adjoint action of \(G\) on \(\mathfrak{g}\). We denote by \(\left\langle\cdot,\cdot\right\rangle\) a Riemannian metric on \(G\). The metric is said to be left-invariant if \(\left\langle u,v\right\rangle_{y}=\left\langle(dL_{x})_{y}u,(dL_{x})_{y}v \right\rangle_{L_{xy}(y)}\), for every \(u,v\in T_{y}G\), i.e., the left-multiplication maps are isometries, for every \(x\in G\). The metric is \(\operatorname{Ad}(G)\)-invariant if \(\left\langle u,v\right\rangle_{e}=\left\langle\operatorname{Ad}(x)u, \operatorname{Ad}(x)v\right\rangle_{e}\), for every \(u,v\in\mathfrak{g}\). Note that an \(\operatorname{Ad}(G)\)-invariant metric on \(G\) is equivalent to a bi-invariant (left- and right-invariant) inner product on \(\mathfrak{g}\). The differential of the \(\operatorname{Ad}\) map at the identity yields a linear map \(\operatorname{ad}(x)=\frac{d}{dt}\operatorname{Ad}(\exp(tx))|_{0}\). This linear map is equal to the Lie bracket \([v,w]=\operatorname{ad}(v)w,v,w\in\mathfrak{g}\). A one-parameter subgroup of \(G\) is a continuous homomorphism \(\gamma\colon(\mathbb{R}_{+})\to G\). The Lie group exponential map \(\exp\colon\mathfrak{g}\to G\) is defined as \(\exp(v)=\gamma_{v}(1)\), for \(v\in\mathfrak{g}\), where \(\gamma_{v}\) is the unique one-parameter subgroup of \(G\) whose tangent vector at \(e\) is \(v\). For matrix Lie groups, the exponential map has the particular form: \(\exp(A)=\sum_{k=0}^{\infty}A^{k}/k!\), for a square matrix \(A\). The resulting matrix \(\exp(A)\) is an invertible matrix. Given an invertible matrix \(B\), if there exists a square matrix \(A\) such that \(B=\exp(A)\), then \(A\) is said to be the logarithm of \(B\). In general, the logarithm might not exist, and if it does, it may fail to be unique. In a neighborhood sufficiently close to the identity, the Lie group logarithm exists and is unique. By means of left-translation (or right-translation), the Lie group exponential map can be extended to a map \(\exp_{\mathbb{S}}\colon T_{g}G\to G\), for all \(g\in G\), defined by \(\exp_{g}(v)=g\exp(dL_{\mathbb{S}^{-1}}v)\). Similarly, the Lie group logarithm at \(g\) becomes \(\log_{g}(v)=dL_{g}\log(g^{-1}v)\). The matrix exponential and logarithms can be computed numerically efficiently (see [23], Chapter 5, and the references therein). **Example 1**.