Hi guys - long time reader first time poster!(adsbygoogle = window.adsbygoogle || []).push({});

I'm currently getting to grips with the topic of Lie Algebras, and I've come across something that's baffled me somewhat. I've been asked to show:

[tex]so(4) = su(2) \oplus su(2)[/tex]

Where the lower so(n) denotes the Lie Algebra of SO(n) etc. Now, in a previous question, I was asked to show:

[tex]u(2) = su(2) \oplus \mathbb{R} [/tex]

Where [tex]\mathbb{R}[/tex] denotes the set of constant (real) multiples of the matrices of the form [tex]i\mathbb{I}_2[/tex]. This was easy enough; I showed that for each [tex]v\in u(2)[/tex] there exists a [tex]x \in su(2)[/tex] and a [tex]y \in \mathbb{R}[/tex] such that v=x+y, and also that if [tex]A \in su(2) \cap \mathbb{R} [/tex] then A=0.

However, in this new case, I assume the [tex]\oplus[/tex] means the the matrix direct sum, but surely if this is the case, then it's false - since the matrices in so(4) take the form:

[tex]\left(\begin{array}{cccc}0&a&b&c\\-a&0&d&e\\-b&-d&0&f\\-c&-e&-f&0\end{array}\right)[/tex]

Which is not of the form of a matrix direct sum. If anyone could give me any hints as to where my confusion lies, I'd be very greatful - although if you could keep the hints sufficiently vague, as to not to do all the work for me!

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# Decomposition of Lie Algebras

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