I have a Lie group $\,\big(\,G,\,\circ\,\big)\,$ of $\,\mathbb{R}^3$ where the law $\,\circ\,$ is defined by
$$ \big(x,\,y,\,z\big)\circ\big(x',\,y',\,z'\big) = \left(x+x',\,z+z'+xy',\,y+y'\right) $$
We denote the left-translation by $\,L_g: p \rightarrow g\circ p$.
Considering a tangent vector $\,v=\left(v_1,\,v_2,\,v_3\right)\,$ of $\,T_xG$, we have $\left(dL_g\right)_e\left(v\right) = \left(v_1,\,v_3,\,v_2\right)\,$ where $\,e=\left(0,\,0,\,0\right)\,$ is the unit element of $\,G$.
I am looking for left invariant vector fields $\,X\left(p\right)=\big(a\left(p\right),b\left(p\right),c\left(p\right)\big)\,$ where $\,a,\,b,\,c\,$ are smooth functions of the coordinates $p=\left(p_1,\,p_2,\,p_3\right)$.
We want $\left(a,b,c\right)$ such that
$\left(dL_g\right)_e\big(X\left(p\right)\big) = X\big(L_g\left(p\right)\big)$
- $X\big(L_g\left(p\right)\big) = \big(a\left(\tilde p\right),\,b\left(\tilde p\right),\,c\left(\tilde p\right)\big)$ and $\,\tilde p = \big(p_1 + g_1,\,g_3+p_3+g_1p_2,\,p_2+g_2\big)$
- $\left(dL_g\right)_e\big(X\left(p\right)\big) = \big(a\left(p\right),\,c\left(p\right),\,b\left(p\right)\big)$
... but I get stuck