### The given information

1. $L \cong L \oplus L$, in a very nice way
2. $m \cong m \oplus m$, in a very nice way
3. $L\cong m \oplus Q_m$, in a nice way
4. $m \cong L \oplus Q_L$, in a faintly dodgy way

### Wizardry from Warsaw

$m \cong L\oplus Q_L \cong (L\oplus L) \oplus Q_L \cong L \oplus (L\oplus Q_L) \cong L \oplus m$

$L \cong m\oplus Q_m \cong (m\oplus m) \oplus Q_m \cong m\oplus (m\oplus Q_m) \cong m\oplus L$

Conclusion: $m \cong L\oplus m \cong m\oplus L \cong L$.

### A nagging question

Just how nice or dodgy is our final isomorphism $m\cong L$?