Let \(G\) be a compact connected simple Lie group with simple Lie algebra \(g\), \(\theta\) and \(\tau\) are two involutions of \(G\) such that \(\theta \tau=\tau \theta\). Let \(K=\{X\in G\,,\,\theta(X)=X\}\), \(K^{\prime}=\{X\in G\), \(\tau(X)=X\}\), and \(K^+=K\cap K^{\prime}\). The authors search for Einstein metrics in a certain \(3\)-parameter family of \(G\)-invariant Riemannian metrics on the homogeneous space \(G/K^+\). A necessary and sufficient condition for a metric from this family to be Einstein is given (theorem 2.1). Finally, some new examples of homogeneous Einstein manifolds with motion group \(G= \text{SU}(n)\) are obtained.