  We derive a new stabilized symmetric Nitsche method for enforcement of
Dirichlet boundary conditions for elliptic problems of second order in cut
isogeometric analysis (CutIGA). We consider $C^1$ splines and stabilize the
standard Nitsche method by adding certain elementwise least squares terms in
the vicinity of the Dirichlet boundary and an additional term on the boundary
which involves the tangential gradient. We show coercivity with respect to the
energy norm for functions in $H^2(\Omega)$ and optimal order a priori error
estimates in the energy and $L^2$ norms. To obtain a well posed linear system
of equations we combine our formulation with basis function removal which
essentially eliminates basis functions with sufficiently small intersection
with $\Omega$. The upshot of the formulation is that only elementwise
stabilization is added in contrast to standard procedures based on ghost
penalty and related techniques and that the stabilization is consistent. In our
numerical experiments we see that the method works remarkably well in even
extreme cut situations using a Nitsche parameter of moderate size.
