Denote by $λK_v$ the complete graph of order $v$ with multiplicity $λ$. Let $λK_v-λK_w-λK_u$ be the graph obtained from $λK_v$ by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity $ λ$ of orders $ w $ and $ u $, respectively. When $λ$ is odd, it is shown that there exists a triangle decomposition of $λK_v-λK_w-λK_u$ if and only if $v\geq w+u+\max\{u,w\}$, $ λ\left({v\choose 2}-{u\choose 2}-{w\choose 2}\right) \equiv 0 \pmod 3$ and $λ(v-w) \equiv λ(v-u) \equiv λ(v-1) \equiv 0 \pmod 2$. When $λ$ is even, it is shown that for large enough $v$, the elementary necessary conditions for the existence of a triangle decomposition of $λK_v-λK_w-λK_u$ are also sufficient.