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For small values of $l$, isomorphisms occur among certain of the classical algebras. Show that $B_2$ is isomorphic to $C_2$.

Well, both $B_2$ and $C_2$ have dimension $10$. $B_2$ consists of $5\times 5$ matrices, while $C_2$ consists of $4\times 4$ matrices. It seems to take too much work to exhibit an explicit isomorphism, and then computing to show that $\phi([x,y])=[\phi(x),\phi(y)]$, where $[x,y]=xy-yx$. How can we show that the two algebras are isomorphic?

PJ Miller
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  • if you have access to a computer algebra system, you can just pick your bases and compute the various commutators. then you can try to see how to cook up a lie algebra isomorphism – citedcorpse Aug 11 '13 at 05:08
  • @exitingcorpse I don't have a computer algebra system. Is there a way to show the isomorphism by hand? – PJ Miller Aug 11 '13 at 05:14
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    Please check the answer in here for a basis-independent proof. – Avitus Aug 18 '13 at 12:38

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