In season 2, episode 8 of the Mentalist, it is revealed that Rebecca Anderson, a secretary in CBI is a Red John operative. She kills a number of CBI agents, etc. After Patrick questions her, she is escorted to a holding cell. On the way there, she is poisoned and dies. The person who poisons her walks right past the officer escorting her to the holding cell.
Compare this with Todd Johnson's murder. He was discovered to be a cop killer (just like Anderson). Patrick refuses to question him, even though Johnson requests to talk to Patrick. Patrick eventually goes to talk to Johnson and finds him on fire in the interrogation room. In the hospital, Johnson's last words to Patrick were "Tyger, Tyger", revealing to Patrick that he too was a Red John operative.
Now, compare the CBI response to both of these killings. With Anderson, they knew she was a Red John operative and she was killed in their custody. CBI does nothing to investigate her death that we are aware of. With Johnson, only Patrick knows. CBI puts JJ LaRoche in charge of investigating his death since there is a murder in the CBI. This investigation plays a significant role in a number of episodes through out the season. Patrick himself is very interested in knowing who the killer was since that leads him to Red John.
Why did they not investigate who killed Anderson? Why wasn't Patrick interested in knowing who had killed her?
Patrick realized she was poisoned due to the very visible marks on her hands, foaming of the mouth, etc. The guard walking her to the holding cell must have seen who walked past seconds before she dies. Furthermore, they could have tested her to figure out that the poison was fast acting, meaning she didn't poison herself hours before, etc.