The length and depth of real algebraic groups

Let $G$ be a connected real algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G=G_0>G_1>...>G_t=1$ where each $G_i$ is a maximal connected real subgroup of $G_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $G$. We give a precise formula for the length of $G$, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If $G$ is simple then we bound the depth of $G$ above and below, and in many cases we compute the exact value. In particular, the depth of any simple $G$ is at most $9$.