both of Lie algebras of type $B_2$ and $C_2$ have dimension 10 and we can find two basis of them on page 3 in the book: Introduction to Lie algebras and representation theory . How could we show that these two Lie algebras are isomorphic by constructing an explicit correspondence between two basis of them. Thank you very much.
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$B_2=so(5)$ and $C_2=sp(4)$. Let me write it as $C_2=sp(V)$ where $V$ is a 4-dim. symplectic vector space with sympl. form $\omega$. The 6-dim. vect. space $\wedge^2 V^*$ has a natural inner product given by the wedge product: $(\alpha,\beta)$ is defined by $\alpha\wedge\beta=(\alpha,\beta)\,\omega\wedge\omega$. Let $W\subset \wedge^2 V^*$ be the orthogonal complement of $\omega\in\wedge^2 V^*$. $W$ is 5-dimensional and has an inner product. The action of $sp(V)$ on $V$ yields an action on $W$ preserving the inner product, i.e. to a Lie algebra morphism $sp(V)\to so(W)$. And this morphism is in fact an isomorphism.