Complex Spin group and covering

Hi, I have the following problem.

To me it appears there is a contradiction between the book
'Supersymmetry for mathematicians' by Varadarajan and the book Spin
Geometry by Lawson & Michelsohn. Varadarajan claims (page 193
http://books.google.nl/books?id=sZ1-G4hQgIIC&printsec=frontcover&dq=supersymmetry+mathematicians&sig=Y1dGVGNKpZUJIjE8eSram6W_09U#PPA193,M1)
that the Spin group for complex vector space V is a double cover of
SO(V).

In the book by Lawson it is claimed (formula 2.28) that it is a 4-
sheeted cover. See http://books.google.nl/books?id=3d9JkN8w3X8C&pg=PP1&dq=spin+geometry&sig=lzJpFgpXK3fHkA2mAJyIA8SrDJQ.

My question is now, who is right? (Or, something that's perhaps more
probable, why am I wrong in seeing a contradiction?)

I hope anyone has the time to help me out with this problem. Thanks in
anticipation!

Seldon

.