Explore the third lecture in a series on Lie algebroid theory, focusing on the concept of ideals in Lie algebroids and their role in geometric reduction. Delve into how ideals function as subrepresentations up to homotopy of adjoint representations, and examine the various obstructions that can arise in Lie algebroids. Learn from research conducted in collaboration with Thiago Drummond and Cristian Ortiz, building upon previous lectures that covered Lie algebroids, their representations up to homotopy, twisted cohomology, and Pontryagin characters of graded vector bundles. Part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" at the Erwin Schrödinger International Institute for Mathematics and Physics.