1. Where has the सूत्रम् 5-4-154 शेषाद्विभाषा (used in step 5) been used in verses 25-30 of Chapter Eighteen of the गीता?
Answer: The सूत्रम् 5-4-154 शेषाद्विभाषा has been used in the form हिंसात्मकः (प्रातिपदिकम् ‘हिंसात्मक’, पुंलिङ्गे प्रथमा-एकवचनम्) in the following verse of the गीता –

रागी कर्मफलप्रेप्सुर्लुब्धो हिंसात्मकोऽशुचिः ।
हर्षशोकान्वितः कर्ता राजसः परिकीर्तितः ॥ 18-27 ॥

हिंसा आत्मा यस्य सः = हिंसात्मकः – one who has an oppressive nature

हिंसा सुँ + आत्मन् सुँ । By 2-2-24 अनेकमन्यपदार्थे।

‘हिंसा सुँ’ – which is qualifying ‘आत्मन् सुँ’ – is placed in the prior position in the compound as per the सूत्रम् 2-2-35 सप्तमीविशेषणे बहुव्रीहौ।
Note: ‘हिंसा सुँ + आत्मन् सुँ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 to apply in the next step.
= हिंसा + आत्मन् । By 2-4-71 सुपो धातुप्रातिपदिकयोः।
= हिंसात्मन् । By 6-1-101 अकः सवर्णे दीर्घः।
= हिंसात्मन् कप् । By 5-4-154 शेषाद्विभाषा।
= हिंसात्मन् क । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम् and 1-3-9 तस्य लोपः।
Note: ‘हिंसात्मन्’ gets the designation पदम् here as per 1-4-17 स्वादिष्वसर्वनामस्थाने। This allows the सूत्रम् 8-2-7 to apply in the next step.
= हिंसात्मक । By 8-2-7 नलोप: प्रातिपदिकान्तस्य।

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example, हिंसात्मकः is qualifying कर्ता। Hence we assign the masculine gender to the compound प्रातिपदिकम् ‘हिंसात्मक’। It declines like राम-शब्द:। प्रथमा-एकवचनम् is हिंसात्मकः।

2. Which other (besides अग्निसाक्षिक:) compound used in the verses is a बहुव्रीहि: compound?
Answer: Besides अग्निसाक्षिकः, the compound सत्यसङ्गरः (प्रातिपदिकम् ‘सत्यसङ्गर’, पुंलिङ्गे प्रथमा-एकवचनम्) is also a बहुव्रीहि: compound.

सत्यः सङ्गरो यस्य सः = सत्यसङ्गरः – one whose promise is true

The अलौकिक-विग्रह: is –
सत्य सुँ + सङ्गर सुँ । By 2-2-24 अनेकमन्यपदार्थे।
The adjective ‘सत्य सुँ’ is placed in the prior position in the compound as per the सूत्रम् 2-2-35 सप्तमीविशेषणे बहुव्रीहौ।
Note: ‘सत्य सुँ + सङ्गर सुँ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 to apply in the next step.
= सत्य + सङ्गर । By 2-4-71 सुपो धातुप्रातिपदिकयोः।
= सत्यसङ्गर ।

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example, सत्यसङ्गरः is qualifying मैथिलः (the king of Mithilā.) Hence we assign the masculine gender to the compound प्रातिपदिकम् ‘सत्यसङ्गर’। It declines like राम-शब्द:। प्रथमा-एकवचनम् is सत्यसङ्गरः।

3. Commenting on the word शेष: used in the सूत्रम् 5-4-154 शेषाद्विभाषा the सिद्धान्तकौमुदी says – शेषशब्दोऽर्थद्वयपर:। Please explain.
Answer: The term ‘शेष’ in the सूत्रम् 5-4-154 शेषाद्विभाषा conveys two meanings.
A बहुव्रीहिसमासः can optionally take the समासान्त: affix कप् (prescribed by 5-4-154) when the compound is –
i) अनुक्तसमासान्तः – no other समासान्त: operation has been prescribed to the compound.
ii) शेषाधिकारस्थः – the compound is composed by a rule in the अधिकारः of शेषः (which runs from 2-2-23 शेषो बहुव्रीहिः to 2-2-24 अनेकमन्यपदार्थे।)

4. Can you spot the negation particle नञ् in the verses?
Answer: The negation particle नञ् occurs in the form अयोनिजाम् (स्त्रीलिङ्ग-प्रातिपदिकम् ‘अयोनिजा’, द्वितीया-एकवचनम्)।

The लौकिक-विग्रहः is –
न योनिजा = अयोनिजा – not born from the womb

Here the negation particle नञ्‌ conveys the sense of विरोध: – opposite of that which is being negated.
The derivation of the compound प्रातिपदिकम् ‘अयोनिज’ is similar to the derivation of the compound प्रातिपदिकम् ‘अपर्याप्त’ shown in answer to question 1 in the following comment – https://avg-sanskrit.org/2015/08/10/अब्राह्मणः-mns/#comment-35496

As per the सूत्रम् 2-4-26 परवल्लिङ्गं द्वन्द्वतत्पुरुषयोः, the compound प्रातिपदिकम् ‘अयोनिज’ is an adjective since the final member ‘योनिज’ is an adjective. In the present example, it is qualifying the feminine noun ‘तनया’। Hence we have to add the appropriate feminine affix (स्त्रीप्रत्यय:) to the प्रातिपदिकम् ‘अयोनिज’।
अयोनिज + टाप् । By 4-1-4 अजाद्यतष्टाप्‌।
= अयोनिज + आ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-7 चुटू, 1-3-9 तस्य लोपः।
= अयोनिजा । By 6-1-101 अकः सवर्णे दीर्घः।
The compound declines like सीता-शब्द:। द्वितीया-एकवचनम् is अयोनिजाम्।

Note: The प्रातिपदिकम् ‘योनिज’ is itself a compound, derived as follows –
योनेर्जातः = योनिजः – born from the womb

’ज’ is a कृदन्त-प्रातिपदिकम् (participle form) derived from the verbal root √जन् (जनीँ प्रादुर्भावे ४. ४४).
The (compound) प्रातिपदिकम् ’योनिज’ is derived as follows:
योनि + ङसिँ + जन् + ड । By 3-2-98 पञ्चम्यामजातौ।
Note: In the सूत्रम् 3-2-98, the term पञ्चम्याम् ends in the seventh (locative) case. Hence ’योनि + ङसिँ’ gets the उपपद-सञ्ज्ञा here by 3-1-92 तत्रोपपदं सप्तमीस्थम्‌।
= योनि + ङसिँ + जन् + अ । अनुबन्ध-लोप: by 1-3-7 चुटू and 1-3-9 तस्य लोपः।
= योनि + ङसिँ + ज् + अ । By 6-4-143 टेः। Note: डित्वसामर्थ्यादभस्यापि टेर्लोपः – The अङ्गम् does not have the भ-सञ्ज्ञा here, but still टि-लोप: is done because otherwise no purpose would be served by having the letter ‘ड’ as a इत् in the affix ’ड’।
= योनि + ङसिँ + ज ।
Now we form a compound between ‘योनि + ङसिँ’ (which is the उपपदम्) and ‘ज’ by using the सूत्रम् 2-2-19 उपपदमतिङ्।
In the compound, ‘योनि + ङसिँ’ is placed in the prior position as per 2-2-30 उपसर्जनं पूर्वम्‌। Note: Here ‘योनि + ङसिँ’ is the उपपदम् and hence it gets the उपसर्जन-सञ्ज्ञा by 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् because in the सूत्रम् 2-2-19 उपपदमतिङ् (which prescribes the compounding) the term उपपदम् ends in the nominative case.
’योनि + ङसिँ + ज’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 to apply next.
= योनि + ज । By 2-4-71 सुपो धातुप्रातिपदिकयोः।
= योनिज ।

5. In which word in the commentary has the सूत्रम् 7-4-46 दो दद् घोः been used?
Answer: The सूत्रम् 7-4-46 दो दद् घोः has been used in the form दत्तवान् (प्रातिपदिकम् ‘दत्तवत्’, पुंलिङ्गे प्रथमा-एकवचनम्) – derived from the verbal root √दा [डुदाञ् दाने ३. १०].
As per 7-4-46 दो दद् घोः – The term ‘दथ्’ is substituted in place of the (entire) verbal root √दा which has the घु-सञ्ज्ञा (ref. 1-1-20), when followed by an affix which is कित् (has letter ‘क्’ as a इत्) and begins with the letter ‘त्’। The entire term ‘दा’ is replaced by ‘दथ्’ as per the सूत्रम् 1-1-55 अनेकाल्शित्सर्वस्य।

Please see the answer to question 5 in the following comment for the derivation of the form दत्तवान् – http://avg-sanskrit.org/2013/05/31/परिच्छदान्-map/#comment-29913

6. How would you say this in Sanskrit?
“This entire universe has the Self as its witness.” Paraphrase to – “This entire universe is that which has the Self as its witness.” Construct a बहुव्रीहि: compound for ‘that which has the Self as its witness’ = ‘आत्मा साक्षी यस्य तत्’।
Answer: इदम् सर्वम् विश्वम् आत्मसाक्षिकम् (अस्ति) = इदं सर्वं विश्वमात्मसाक्षिकम्।
अथवा –
इदम् सर्वम् विश्वम् आत्मसाक्षि (अस्ति) = इदं सर्वं विश्वमात्मसाक्षि।

Easy Questions:
1. Where has the सूत्रम् 7-3-102 सुपि च been used in the verses?
Answer: The सूत्रम् 7-3-102 सुपि च has been used in the form राघवाय (पुंलिङ्ग-प्रातिपदिकम् ‘राघव’, चतुर्थी-एकवचनम्)।
राघव + ङे । By 4-1-2 स्वौजसमौट्छष्टाभ्याम्भिस्ङेभ्याम्भ्यस्ङसिभ्याम्भ्यस्ङसोसाम्ङ्योस्सुप्‌।
= राघव + य । By 7-1-13 ङेर्यः।
= राघवाय । By 7-3-102 सुपि च – the ending letter ‘अ’ of a प्रातिपदिकम् is elongated if it is followed by a सुँप् affix beginning with a letter of the यञ्-प्रत्याहार:।

2. Which सूत्रम् prescribes the substitution ‘औ’ in the form संनिधौ (पुंलिङ्ग-प्रातिपदिकम् ‘संनिधि’, सप्तमी-एकवचनम्)?
Answer: The सूत्रम् 7-3-119 अच्च घेः prescribes the substitution ‘औ’ in the form संनिधौ (पुंलिङ्ग-प्रातिपदिकम् ‘संनिधि’, सप्तमी-एकवचनम्)।

संनिधि + ङि । By 4-1-2 स्वौजसमौट्छष्टाभ्याम्भिस्ङेभ्याम्भ्यस्ङसिभ्याम्भ्यस्ङसोसाम्ङ्योस्सुप्‌। ‘संनिधि’ has the घि-सञ्ज्ञा by 1-4-7 शेषो घ्यसखि। This allows 7-3-119 to apply in the next step.
= संनिध + औ । By 7-3-119 अच्च घे: – Following a short ‘इ’ or short ‘उ’ ending अङ्गम्, the affix ‘ङि’ is replaced by ‘औ’ and the (ending letter ‘इ’ or ‘उ’ of the) अङ्गम् which has the घि-सञ्ज्ञा is replaced by short ‘अ’।
= संनिधौ । By 6-1-88 वृद्धिरेचि।