Final answer:
Double chromosomes prepare for splitting during the S phase of the cell cycle when each chromosome is replicated to form sister chromatids. They then separate during the mitotic phase.
Step-by-step explanation:
The conditions under which double chromosomes are preparing for splitting is during the S phase of the cell cycle. During the S phase, each chromosome is replicated to produce two identical copies, known as sister chromatids. These sister chromatids are held together at the centromere by cohesin proteins.
This replication is essential for cells to have the correct number of chromosomes after cell division. The mitotic phase, consisting of prophase, metaphase, anaphase, and telophase, follows interphase and is where the actual separation of sister chromatids occurs.