Ikeda type lift on ${\rm SO}(3,n+1)$

By using Ikeda's theory for a compatible family of Eisenstein series, we explicitly construct Ikeda type lifts on the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$ with $n\ge 3$ which splits everywhere at finite places. Our lifts are Hecke eigen cusp forms of weight $l$ ($l\ge n+2$, even) and come from elliptic newforms with respect to ${\rm SL}_2(\mathbb{Z})$ which are of weight $l-\frac{n-2}{2}$ when $n$ is even and $2l-n+1$ when $n$ is odd.The corresponding cuspidal automorphic representations are cohomological but non-tempered at any places including the infinity place.