On the embeddability of $[3]*K$

We relate the embeddability of the simplicial complex $[3]*K$ into $\mathbb{R}^{n+2}$ to that of $K$ into $\mathbb{R}^n$. In brief, the embeddability of $K$ into $\mathbb{R}^n$, in the metastable range $2n\geq 3(d+1)$, is equivalent to the embeddability of $[3]*K$ into $\mathbb{R}^{n+2}$. We show moreover than the van Kampen obstruction of $K$ vanishes if and only if the van Kampen obstruction of $[3]*K$ vanishes. It follows that for $d=2$, embeddabilty of $[3]*K$ is equivalent with the vanishing of the van Kampen obstruction for $K$, but not with the embeddability of $K$.