# Table 2 Elements of the Lie algebra $$\tilde{\mathfrak g}$$ and the associated Lie group $$\tilde{G}$$
Element $$\varvec{{\tilde{\mathbf {Z}}}}_{\varvec{I,r}} \varvec{\in \tilde{\mathfrak g}}$$ Element $${\varvec{\exp \left( {\tilde{{\mathbf {Z}}}_{I,r}}\right) \in \tilde{G}}}$$ Subalgebra in $$\varvec{\tilde{\mathfrak g}}$$
$$\tilde{\mathbf {Z}}_{1,r} := \begin{pmatrix} z_{1,r} &{} -z_{1,r} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}$$ $$\exp \left( {\tilde{\mathbf {Z}}_{1,r}}\right) = \begin{pmatrix} \exp \left( {z_{1,r}}\right) &{} 1- \exp \left( {z_{1,r}}\right) &{} 0\\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}$$ $$\tilde{\mathfrak g}_1 = \langle \tilde{\mathbf {Z}}_{1,r} \rangle$$, $$r \in \{1,2\}$$
$$\tilde{\mathbf {Z}}_{2,r} := \begin{pmatrix} z_{2,r} &{} 0 &{} -z_{2,r} \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\end{pmatrix}$$ $$\exp \left( {\tilde{\mathbf {Z}}_{2,r}}\right) = \begin{pmatrix} \exp \left( {z_{2,r}}\right) &{} 0 &{} 1-\exp \left( {z_{2,r}}\right) \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}$$ $$\tilde{\mathfrak g}_2 = \langle \tilde{\mathbf {Z}}_{2,r} \rangle$$, $$r \in \{1,2\}$$
1. $$z_{I,r}$$ assumes the following values: $$z_{1,1} = z_{2,1} = 1$$, $$z_{1,2} = z_{2,2} = \mathrm{i}$$