Definition of a algebra generated by a set ?

Hello,

I'm reading a book about mathematical physics and have a lot of
troubles understanding some examples in this book.
First they define a algebra as vector space A over a field K with a
product . defined that satisfies A.(bB+cC)=bA.B+cA.C and (bB+cC).A=bB.A
+cC.A, the algebra is commutative (associative) if the product is
associative (commutative). No problem here. However at one moment they
start to give examples like:
Let Q be the associative algebra over R generated by the four elements
{1,i,j,k} satisfying i^2=j^2=k^2=-1, ij=k,jk=i,ki=j, 1^2=1,1i=i,1j=j,1k=k
which leads to the quaternions.
Let V be a real vector space with the inner product u.v and e1,...en a
orthonormal basis, the Clifford algebra associated with the inner product
space is the associative algebra generated by 1,e1,..en with product
rules eiej+ejei=2gij 1 (gij=ei.ej)
Now I understand that you can create the quaternions as a set of tuples
of real numbers with the correct multiplication, but I fail to see how
you can take just 4 elements {1,i,j,k} and then generate a vector space
over a field together with a product (unless you define 1=(1,0,0,0) , i=
(0,1,0,0), j=(0,0,1,0) and k=(0,0,0,1) and define addition in the usual
way (product is of course a little bit more complex), but you can hardly
call this generating.
Also when I look up Clifford algebras on the internet, It is never
defined as the associative algebra generated by ....

There seems to be a pattern here where given a set of undefined elements
{1,a1,...an) you go on and generate a algebra out of this. So how do you
proceed to define a vector space (definition of the sum, multiplication
by a scalar (what is the field used) and the algebra product. I looked
really hard but could not find such a definition in the book neither in
other books I have on algebra neither on the internet. Either I'm missing
something obvious or my book is based on some loosy mathematical
principles.

Thanks a lot for any help


Marc Mertens
.

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