division algebras - Why don't quaternions have an overall phase?

There is a unique copy of $mathbb R$ in $mathbb H$ and it is the center.
On the other hand, for every quaternion $q$ not in $mathbb R$, the generated $mathbb R$-algebra with unit is a copy of $mathbb C$, but none is central.

I like to view quaternions as $q=(x,X)in mathbb Rtimes mathbb R^3$ with multiplication
$$
(x,X).(y,Y) = (x.y - langle X,Yrangle, X times Y + x.Y + y.X)
$$
This uses oriented Euclidean $mathbb R^3$, and it turns
http://en.wikipedia.org/wiki/Quaternion#Quaternions_and_the_geometry_of_R3
into a definition.
One can then take any oriented orthonormal basis $i,j,k$ for the basic imaginary quaternions.

Edit: Note that $Xtimes Y$ is a Lie bracket and $-langle X,Yrangle$ the corresponding Killing form ($mathfrak smathfrak o(3,mathbb R)$ up to a constant), one can repeat this construction for every real Lie algebra. Only for 3 of them one obtains an associative algebra.

Second edit: The reference for the first edit is:

Peter W. Michor, Wolfgang Ruppert, Klaus Wegenkittl: A connection between Lie algebras and general algebras. Rendiconti Circolo Matematico di Palermo, Serie II, Suppl. 21 265--274, (1989)(pdf)

Moreover, a similar construction as the above on $mathbb Ctimes mathbb C^3$ using also complex conjugation (see "Greub: Multilinear algebra, 2n ed. 1978" page 289) leads to octonians.