This is my first post here so I apologise in advance if I've done anything against the rules. My question is with regards to an answer given here Exercise 7.7.3 in Weibel (computation of $H^3(\mathfrak{sl}_2,k)$ via Chevalley-Eilenberg complex). For context I have no more than a basic undergrad grounding in homological algebra.
In the answer given by the user "m_t_" I can follow every step of the argument up until the conclusion, specifically why the map being zero implies the final result.
In the context of homological algebra I'm most comfortable with, I attempted the question myself. Consider the chain complex; $$ 0 \overset{d_{0}}{\leftarrow} \Lambda^{0}(\mathfrak{sl}_{2}) \overset{d_{1}}{\leftarrow} \Lambda^{1}(\mathfrak{sl}_{2}) \overset{d_{2}}{\leftarrow} \Lambda^{2}(\mathfrak{sl}_{2}) \overset{d_{3}}{\leftarrow} \Lambda^{3}(\mathfrak{sl}_{2}) \overset{d_{4}}{\leftarrow}0 $$
Due to the fact that $ d_{3}(f \wedge h \wedge e) = 0 $, I obtain $ \ker(d_{3}) = \Lambda^{3}(\mathfrak{sl}_{2}) $ and $ \text{im}(d_{4}) = 0$. Which implies $ H_{3}(\mathfrak{sl}_{2}) = \Lambda^{3}(\mathfrak{sl}_{2}) $. But I don't believe that is correct. Where exactly has my computation or understanding gone wrong? Any help or even simple recources would be appreciated.