K3 surfaces with two involutions and low Picard number

Let X be a complex algebraic K3 surface of degree 2d and with Picard number rho. Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, rho >= 1 when d = 1 and rho >= 2 when d >= 2. For d = 1, the first example defined over Q with rho = 1 was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondo, also defined over Q, can be used to realise the minimum rho = 2 for all d >= 2. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum rho = 2 for d = 2, 3, 4. We also show that a nodal quartic surface can be used to realise the minimum rho = 2 for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank 1 <= r <= 10 and signature (1, r - 1) there exists a K3 surface Y defined over R such that Pic Y-C = Pic Y congruent to N.