Definition 5

A Lie group germ over K is a system $(G, e, \theta, m)$ satisfying the following conditions: (i) $G$ is an analytic manifold over $K$; (ii) $e \in G$; (iii) $\theta$ is an analytic mapping of $G$ into $G$; (iv) $m$ is an analytic mapping of an open subset $\Omega$ of $G \times G$ into $G$; (v) for all $g \in G$, $(e, g) \in \Omega$, $(g, e) \in \Omega$, $m(e, g) = m(g, e) = g$; (vi) for all $g \in G$, $(g, \theta(g)) \in \Omega$, $(\theta(g), g) \in \Omega$, $m(g, \theta(g)) = m(\theta(g), g) = e$; (vii) if $g, h, k$ are elements of $G$ such that $(g, h) \in \Omega$, $(h, k) \in \Omega$, $(m(g, h), k) \in \Omega$, $(g, m(h, k)) \in \Omega$, then $m(m(g, h), k) = m(g, m(h, k))$.

$e$ is called the identity element of the group germ. We often write $gh$ instead of $m(g, h)$ and (by an abuse of notation) $g^{-1}$ instead of $\theta(g)$.

A Lie group $G$ is a Lie group germ with the obvious choice of $e, \theta, m$.

Let $G$ be a Lie group germ. Then $ee^{-1} = e$, that is

For all $g \in G$,

that is

A subset of $G$ invariant under the mapping $g \mapsto g^{-1}$ is called symmetric.

The manifold $G$, with the point $e$, the mapping $g \mapsto g^{-1}$ and the mapping $(g, h) \mapsto hg$ is a Lie Group germ $G^\vee$ called the opposite of $G$.

The Lie group germ $G$ is called commutative if, for all $(g, h) \in G \times G$ such that $gh$ is defined, $hg$ is defined and equal to $gh$.

Let $G$ be a Lie group germ. The set of $(g, h) \in G \times G$ such that $gh$ is defined is a neighbourhood of $(e, e)$. On the other hand, the mappings $(g, h) \mapsto gh$ and $g \mapsto g^{-1}$ are continuous. Hence $(gh)k = g(hk)$ for $g, h, k$ sufficiently close to $e$. Similarly, $(h^{-1}g^{-1})(gh) = h^{-1}(eh) = h^{-1}h = e$ for $g, h$ sufficiently close to $e$, whence multiplying on the right by $(gh)^{-1}$,