Complete intersections of quadrics and the weak Lefschetz property

We consider graded artinian complete intersection algebras A=C[x0,...,xm]/I with I generated by homogeneous forms of degree d2. We show that the general multiplication by a linear form L:Ad-1Ad is injective. We prove that the Weak Lefschetz Property for holds for any c.i. algebra A as above with d=2 and m4, previously known for m <= 3.