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Since every $4×4$ skew-symmetric matrix can be written uniquely as a decomposition
$$\begin{bmatrix} 0&-a&-b&-c \\a&0&-c&b\\b&c&0&-a\\c&-b&a&0\end{bmatrix}+\begin{bmatrix} 0&-x&-y&-z \\x&0&z&-y\\y&-z&0&x\\z&y&-x&0\end{bmatrix}$$ If $E_{ij}=e_{ij}-e_{ji}$ for $i<j$. We can find that if $I = −E_{12} −E_{34}$, $J = −E_{13} +E_{24}$, and $K = −E_{14} −E_{23}$, then $[I,J] = 2K$, $[J,K] = 2I$, and $[K, I] = 2J$.

My question is how can we deduce from what we have above that $\mathfrak {so}(4)\cong \mathfrak {so}(3)\oplus \mathfrak {so}(3) $?

JKnecht
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Ronald
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    Yes, we can specify a basis $(e_1,\ldots ,e_6)$ for $\mathfrak{so}(4)$ this way and then define $\phi(e_i)$ such that $\phi\colon \mathfrak{so}(4)\rightarrow \mathfrak{so}(3)\times \mathfrak{so}(3)$ is a Lie algebra homomorphism with $\det(\phi)\neq 0$. I have answered your question already here. – Dietrich Burde Feb 11 '16 at 09:58
  • Sorry, I don't understand where to use the decomposition, can you please explain it to me? – Ronald Feb 11 '16 at 13:16

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