Solarii Brotherhood

The Solarii Brotherhood are the primary antagonists encountered in Tomb Raider as well as the majority of the humans residing on Yamatai. Very few members of the Solarii were referred to by name, mainly just Father Mathias, their evil master, Vladimir, the leader of a group of scavengers living in a mountainside village, and Vlad's brothers Boris, Nikolai, and Dmitri.

Rise of the Brotherhood
When Mathias arrived on Yamatai, his fellow survivors died trying to escape the island, and he knew there was something more to the storms. As he was discovering the secrets of the island, he found that he could not do it alone, he gathered a small group of loyal followers. Believing they must understand power and the need to organize. But more importantly, they cannot hesitate to kill or use violence, making it their Life there. This began the Brotherhood of Solarii, by giving them structure, purpose and work.

After explaining the truth of the island, he would offer the survivors a choice: death or salvation. Those who opposed would be immediately killed. Women would be sacrificed (or, from a certain point of view, tested) in their ceremony to find the Sun Queen's successor.

Mathias recruited as many as possible, choosing those who are physically strong, but not too smart, so they would fall in line and take orders. Those who asked questions would be killed immediately. He drafted laws and a code for them to live by. They would need to prove their commitment to the queen, making them fall in line and believe.

The Solarii built a city near the Queen's palace while Mathias searched for the key to unlocking her power and escaping the island.

Initiation
As stated by a Solarii in one of the Confessions of a Solarii entries, Mathias believed that the men would need to be strong of body, weak of will. Because of this, the new recruits are thrown into a chamber in the Geothermal Caverns and were left to fend for themselves for days, sometimes weeks, without water or foods. Those who survived are welcomed with open arms. But at the same time, the recruits' humanity was destroyed, turning them into inhumane savages.

The lack of food and water caused most of the recruits to resort to cannibalism, wasting no time tearing apart the new ones and eat them, as witnessed by Lara after escaping from her captors. Some recruits are also locked in cells, their sanity slowly dissolving into madness.

After Himiko's Death
After Lara destroyed Himiko and left the island, alongside Jonah, Reyes and Sam. The Solarii Brotherhood were left mostly unarmed and leaderless. However, not long after Trinity took notice and sent in their Elite in order to wipe out everything there, ranging from cultural artifacts, destroying the tombs, killing and disposing of the Solarii Brotherhood and Stormguard warriors. All this was done before Yamatai gained media attention.

Trivia

 * Lara would find a teddy bear and toy train belonging to little girls during her journey. They were with their father on a day trip on a plane, where it got caught in Himiko's storm and crashed on the island. While exploring the geothermal caverns, you can listen to some of the prisoners' crazy rambling. One will speak of being on a trip with his wife and daughter, implying that they were killed, but he mentions being on a boat, not a plane, so it is unlikely that he is their father.