  We prove that the commutation relations among the generators of the quadratic
Heisenberg algebra of dimension $n\in\mathbb{N}$, look like a kind of
\textit{non-commutative extension} of $\hbox{sl}(2, \mathbb{C})$ (more
precisely of its unique $1$--dimensional central extension), denoted
$\hbox{heis}_{2;\mathbb{C}}(n)$ and called the complex $n$--dimensional
quadratic Boson algebra. This \textit{non-commutativity} has a different nature
from the one considered in quantum groups. %In particular we prove that, for
%most values of $n$, this Lie algebra cannot be isomorphic to %$\hbox{sl}(N,
\mathbb{C})$ for almost any value of $N$. We prove the exponentiability of
these algebras (for any $n$) in the Fock representation. We obtain the group
multiplication law, in coordinates of the first and second kind, for the
quadratic Boson group and we show that, in the case of the adjoint
representation, these multiplication laws can be expressed in terms of a
generalization of the Jordan multiplication. We investigate the connections
between these two types of coordinates (disentangling formulas). From this we
deduce a new proof of the expression of the vacuum characteristic function of
homogeneous quadratic boson fields.
