I'm studying the "Clifford Lie Algebra" (see http://arxiv.org/pdf/1007.2481.pdf page 30). It's basically a way to look at Clifford algebras and their properties in a Lie algebraic setting (which I find appealing). I'm looking for a reference that looks at this in more detail; one that discusses the lie subalgebras even better. Thanks.
A little bit of what you want can be found in Chapter 5 of GraciaBondia, Varilly, and Figueroa's book Elements of Noncommutative Geometry. They don't say much about subalgebras, I think, but they do prove (Lemma 5.7, page 182) the result that bivectors in the Clifford algebra $Cl(V)$ are closed under taking commutators, and that the adjoint action of bivectors on vectors in $V$ induces an isomorphism of the Lie algebra of bivectors with $\mathfrak{so}(V)$. That might be a good place to start. 


I don't know anyone else who calls this the "Clifford Lie Algebra". It is just one of the basic applications of Clifford algebras. Given the Clifford algebra of a quadratic form, the quadratic elements of the Clifford algebra give you the Lie algebra of the orthogonal group of that quadratic form. There are many places to read about this, one of them would be Chapter 1.6 of "Spin Geometry" by Lawson and Michelson. I've written up some notes for a graduate course that include this, see here: http://www.math.columbia.edu/%7Ewoit/LieGroups2012/cliffalgsandspingroups.pdf 

