Definition 1

Let r be an integer $\geq 0$. A standard group of dimension r over K is a Lie group G with the following properties: (i) the underlying analytic manifold of G is $m \times m \times \cdots \times m$ (r factors); (ii) there exists a formal power series F in 2r variables with coefficients in $A^r$, without constant term, such that $x.y = F(x, y)$ for all $x, y$ in G.

Then $0.0 = 0$ and hence the identity element of G is the origin of $m \times m \cdots \times m$.

† The results of nos. 3 and 4 and their proofs remain true when the characteristic of K is $> 0$.

L(G) will be identified with K'. By § 5, formula (13), the constants of structure of L(G) with respect to the canonical basis belong to A. We shall need, in the same proof to consider the elements of m × ... × m, now as elements of G, now as elements of L(G).