A cube has vertices $P_1,P_2,P_3,P_4,P_1^{\prime },P_2^{\prime },P_3^{\prime },$ and $P_4^{\prime }$. The side length of the cube is 2 units. Vertices $P_2,P_3,$ and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4$, vertices $P_i$ and $P_i^{\prime }$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1{P}_2,P_1{P}_3,P_1{P}_4,P_1^{\prime }{P}_2^{\prime },P_1^{\prime }{P}_3^{\prime },$ and $P_1^{\prime }{P}_4^{\prime }$. What is the octahedron's volume?