  We prove that the Lawson surface $\xi_{g,1}$ in Lawson's original notation,
which has genus $g$ and can be viewed as a desingularization of two orthogonal
great two-spheres in the round three-sphere ${\mathbb{S}}^3$, has index $2g+3$
and nullity $6$ for any genus $g\ge2$. In particular $\xi_{g,1}$ has no
exceptional Jacobi fields, which means that it cannot `flap its wings' at the
linearized level and is $C^1$-isolated.
