© 2019 American Mathematical Society. If G is a finite group and k is an algebraically closed field of characteristic 2, we show that the number of isomorphism classes of quadratic type principal indecomposable kG-modules is equal to the number of conjugacy classes of strongly real odd order elements of G. If ϕ is a self-dual irreducible 2-Brauer character of G, we show that the corresponding principal indecomposable kG-module has quadratic type if and only if ϕ(g)/2 is not an algebraic integer for some strongly real odd order element g in G.