For fixed \(\lambda >0,\) the polynomials \(\left\{ A_{n} \right\} _{n=0}^{\infty }\) orthogonal with respect to the Sobolev inner product \(\begin{aligned} \langle f,g\rangle _{\lambda }:=\int _{-1}^{1}(f(x)\overline{g}(x)+\lambda f^{\prime }(x)\overline{g}^{\prime }(x))dx \end{aligned}\) are called Althammer’s polynomials or Sobolev-Legendre polynomials. They were first studied in 1962 by P. Althammer. In this paper, for any fixed \(c>0,\) we study the orthogonal polynomials \(\left\{ K_{n}\right\} _{n=0}^{\infty }\) with respect to the (positive-definite) inner product \(\begin{aligned} (f,g)_{1}:=-\frac{\left( f(1)-f(-1)\right) \left( \overline{g} (1)-\overline{g}(-1)\right) }{2}+\int _{-1}^{1}(f^{\prime }(x)\overline{g}^{\prime }(x)+cf(x)\overline{g}(x))dx. \end{aligned}\) We call these polynomials Krein–Sobolev polynomials. The inner product \((\cdot ,\cdot )_{1}\) naturally emerges from a left-definite spectral study of the one-dimensional shifted Krein Laplacian operator \(S_{K}\) . In this paper, we review some properties of \(S_{K}\) and we establish various properties of these Krein–Sobolev polynomials, including completeness as well as the reality and simplicity of their roots in \((-1,1).\)